CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Arithmetic - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"eb5d5097-b0df-4901-8777-0f81bd026f19","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Arithmetic","wgTitle":"Arithmetic","wgCurRevisionId":1272376224,"wgRevisionId":1272376224,"wgArticleId":3118,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Good articles","Pages using multiple image with auto scaled images","CS1 maint: DOI inactive as of December 2024","Arithmetic"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Arithmetic","wgRelevantArticleId":3118,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject": "wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":200000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q11205","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader", "ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.18"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/1200px-Arithmetic_operations.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1200"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/800px-Arithmetic_operations.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="800"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/640px-Arithmetic_operations.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="640"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Arithmetic - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Arithmetic"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Arithmetic&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Arithmetic"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Arithmetic rootpage-Arithmetic skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Arithmetic" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Arithmetic" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Arithmetic" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Arithmetic" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition,_etymology,_and_related_fields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition,_etymology,_and_related_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition, etymology, and related fields</span> </div> </a> <ul id="toc-Definition,_etymology,_and_related_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Numbers</span> </div> </a> <button aria-controls="toc-Numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Numbers subsection</span> </button> <ul id="toc-Numbers-sublist" class="vector-toc-list"> <li id="toc-Kinds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Kinds</span> </div> </a> <ul id="toc-Kinds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numeral_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeral_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Numeral systems</span> </div> </a> <ul id="toc-Numeral_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operations</span> </div> </a> <button aria-controls="toc-Operations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Operations subsection</span> </button> <ul id="toc-Operations-sublist" class="vector-toc-list"> <li id="toc-Addition_and_subtraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition_and_subtraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Addition and subtraction</span> </div> </a> <ul id="toc-Addition_and_subtraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication_and_division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplication_and_division"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Multiplication and division</span> </div> </a> <ul id="toc-Multiplication_and_division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponentiation_and_logarithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponentiation_and_logarithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Exponentiation and logarithm</span> </div> </a> <ul id="toc-Exponentiation_and_logarithm-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Integer_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integer_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Integer arithmetic</span> </div> </a> <ul id="toc-Integer_arithmetic-sublist" class="vector-toc-list"> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Number theory</span> </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rational_number_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_number_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Rational number arithmetic</span> </div> </a> <ul id="toc-Rational_number_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_number_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_number_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Real number arithmetic</span> </div> </a> <ul id="toc-Real_number_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximations_and_errors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximations_and_errors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Approximations and errors</span> </div> </a> <ul id="toc-Approximations_and_errors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tool_use" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tool_use"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Tool use</span> </div> </a> <ul id="toc-Tool_use-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Others" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Others"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Others</span> </div> </a> <ul id="toc-Others-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Axiomatic_foundations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Axiomatic_foundations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Axiomatic foundations</span> </div> </a> <ul id="toc-Axiomatic_foundations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_various_fields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_various_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In various fields</span> </div> </a> <button aria-controls="toc-In_various_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In various fields subsection</span> </button> <ul id="toc-In_various_fields-sublist" class="vector-toc-list"> <li id="toc-Education" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Education"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Education</span> </div> </a> <ul id="toc-Education-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Psychology"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Psychology</span> </div> </a> <ul id="toc-Psychology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philosophy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Philosophy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Philosophy</span> </div> </a> <ul id="toc-Philosophy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Others_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Others_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Others</span> </div> </a> <ul id="toc-Others_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Arithmetic</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 163 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-163" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">163 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Rekenkunde" title="Rekenkunde – Afrikaans" lang="af" hreflang="af" data-title="Rekenkunde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Arithmetik" title="Arithmetik – Alemannic" lang="gsw" hreflang="gsw" data-title="Arithmetik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%8A%90_%E1%89%81%E1%8C%A5%E1%88%AD" title="ሥነ ቁጥር – Amharic" lang="am" hreflang="am" data-title="ሥነ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अंकगणित – Angika" lang="anp" hreflang="anp" data-title="अंकगणित" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ab mw-list-item"><a href="https://ab.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Abkhazian" lang="ab" hreflang="ab" data-title="Арифметика" data-language-autonym="Аԥсшәа" data-language-local-name="Abkhazian" class="interlanguage-link-target"><span>Аԥсшәа</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A%D8%A7%D8%AA" title="حسابيات – Arabic" lang="ar" hreflang="ar" data-title="حسابيات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Aragonese" lang="an" hreflang="an" data-title="Aritmetica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B9%D5%B8%D6%82%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%AB%D6%82%D5%B6" title="Թուաբանութիւն – Western Armenian" lang="hyw" hreflang="hyw" data-title="Թուաբանութիւն" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%9F%E0%A7%80%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="পাটীগণিত – Assamese" lang="as" hreflang="as" data-title="পাটীগণিত" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Asturian" lang="ast" hreflang="ast" data-title="Aritmética" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Hesab" title="Hesab – Azerbaijani" lang="az" hreflang="az" data-title="Hesab" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8" title="حساب – South Azerbaijani" lang="azb" hreflang="azb" data-title="حساب" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%9F%E0%A6%BF%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="পাটিগণিত – Bangla" lang="bn" hreflang="bn" data-title="পাটিগণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/So%C3%A0n-su%CC%8Dt" title="Soàn-su̍t – Minnan" lang="nan" hreflang="nan" data-title="Soàn-su̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ba.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Bashkir" lang="ba" hreflang="ba" data-title="Арифметика" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D1%8B%D1%84%D0%BC%D0%B5%D1%82%D1%8B%D0%BA%D0%B0" title="Арыфметыка – Belarusian" lang="be" hreflang="be" data-title="Арыфметыка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D1%80%D1%8B%D1%82%D0%BC%D1%8D%D1%82%D1%8B%D0%BA%D0%B0" title="Арытмэтыка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Арытмэтыка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अंकगणित – Bhojpuri" lang="bh" hreflang="bh" data-title="अंकगणित" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Central Bikol" lang="bcl" hreflang="bcl" data-title="Aritmetika" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bi mw-list-item"><a href="https://bi.wikipedia.org/wiki/Arithmetic" title="Arithmetic – Bislama" lang="bi" hreflang="bi" data-title="Arithmetic" data-language-autonym="Bislama" data-language-local-name="Bislama" class="interlanguage-link-target"><span>Bislama</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Аритметика – Bulgarian" lang="bg" hreflang="bg" data-title="Аритметика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Bosnian" lang="bs" hreflang="bs" data-title="Aritmetika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Breton" lang="br" hreflang="br" data-title="Aritmetik" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D1%8D" title="Арифметикэ – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Арифметикэ" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aritm%C3%A8tica" title="Aritmètica – Catalan" lang="ca" hreflang="ca" data-title="Aritmètica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Chuvash" lang="cv" hreflang="cv" data-title="Арифметика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Cebuano" lang="ceb" hreflang="ceb" data-title="Aritmetik" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target"><span>Cebuano</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Czech" lang="cs" hreflang="cs" data-title="Aritmetika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Chavacano" lang="cbk" hreflang="cbk" data-title="Aritmetica" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Huwandu" title="Huwandu – Shona" lang="sn" hreflang="sn" data-title="Huwandu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Corsican" lang="co" hreflang="co" data-title="Aritmetica" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhifyddeg" title="Rhifyddeg – Welsh" lang="cy" hreflang="cy" data-title="Rhifyddeg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Danish" lang="da" hreflang="da" data-title="Aritmetik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Arithmetik" title="Arithmetik – German" lang="de" hreflang="de" data-title="Arithmetik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Aritmeetika" title="Aritmeetika – Estonian" lang="et" hreflang="et" data-title="Aritmeetika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%B9%CE%BA%CE%AE" title="Αριθμητική – Greek" lang="el" hreflang="el" data-title="Αριθμητική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Spanish" lang="es" hreflang="es" data-title="Aritmética" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Aritmetiko" title="Aritmetiko – Esperanto" lang="eo" hreflang="eo" data-title="Aritmetiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Extremaduran" lang="ext" hreflang="ext" data-title="Aritmética" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Basque" lang="eu" hreflang="eu" data-title="Aritmetika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8" title="حساب – Persian" lang="fa" hreflang="fa" data-title="حساب" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Arithmetic" title="Arithmetic – Fiji Hindi" lang="hif" hreflang="hif" data-title="Arithmetic" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Arithm%C3%A9tique" title="Arithmétique – French" lang="fr" hreflang="fr" data-title="Arithmétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhr%C3%ADocht" title="Uimhríocht – Irish" lang="ga" hreflang="ga" data-title="Uimhríocht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/%C3%80ireamhachd" title="Àireamhachd – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Àireamhachd" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Galician" lang="gl" hreflang="gl" data-title="Aritmética" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%AE%97%E8%A1%93" title="算術 – Gan" lang="gan" hreflang="gan" data-title="算術" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%85%E0%AA%82%E0%AA%95%E0%AA%97%E0%AA%A3%E0%AA%BF%E0%AA%A4" title="અંકગણિત – Gujarati" lang="gu" hreflang="gu" data-title="અંકગણિત" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-gom mw-list-item"><a href="https://gom.wikipedia.org/wiki/Ank%E2%80%8Cgonnit" title="Ank‌gonnit – Goan Konkani" lang="gom" hreflang="gom" data-title="Ank‌gonnit" data-language-autonym="गोंयची कोंकणी / Gõychi Konknni" data-language-local-name="Goan Konkani" class="interlanguage-link-target"><span>गोंयची कोंकणी / Gõychi Konknni</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Kalmyk" lang="xal" hreflang="xal" data-title="Арифметика" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%B0%EC%88%A0" title="산술 – Korean" lang="ko" hreflang="ko" data-title="산술" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%BE%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Թվաբանություն – Armenian" lang="hy" hreflang="hy" data-title="Թվաբանություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अंकगणित – Hindi" lang="hi" hreflang="hi" data-title="अंकगणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Croatian" lang="hr" hreflang="hr" data-title="Aritmetika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Aritmetiko" title="Aritmetiko – Ido" lang="io" hreflang="io" data-title="Aritmetiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Iloko" lang="ilo" hreflang="ilo" data-title="Aritmetika" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Indonesian" lang="id" hreflang="id" data-title="Aritmetika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Arithmetica" title="Arithmetica – Interlingua" lang="ia" hreflang="ia" data-title="Arithmetica" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%C3%A6" title="Арифметикæ – Ossetic" lang="os" hreflang="os" data-title="Арифметикæ" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-arithmetic" title="I-arithmetic – Xhosa" lang="xh" hreflang="xh" data-title="I-arithmetic" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ubuphicinani" title="Ubuphicinani – Zulu" lang="zu" hreflang="zu" data-title="Ubuphicinani" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Talnareikningur" title="Talnareikningur – Icelandic" lang="is" hreflang="is" data-title="Talnareikningur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Italian" lang="it" hreflang="it" data-title="Aritmetica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="אריתמטיקה – Hebrew" lang="he" hreflang="he" data-title="אריתמטיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Ng%C3%A8lmu_%C3%A9tung" title="Ngèlmu étung – Javanese" lang="jv" hreflang="jv" data-title="Ngèlmu étung" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kl mw-list-item"><a href="https://kl.wikipedia.org/wiki/Aritmetikki" title="Aritmetikki – Kalaallisut" lang="kl" hreflang="kl" data-title="Aritmetikki" data-language-autonym="Kalaallisut" data-language-local-name="Kalaallisut" class="interlanguage-link-target"><span>Kalaallisut</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%82%E0%B2%95%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಅಂಕಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ಅಂಕಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%94%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="არითმეტიკა – Georgian" lang="ka" hreflang="ka" data-title="არითმეტიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Kazakh" lang="kk" hreflang="kk" data-title="Арифметика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niveronieth" title="Niveronieth – Cornish" lang="kw" hreflang="kw" data-title="Niveronieth" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Hesabu" title="Hesabu – Swahili" lang="sw" hreflang="sw" data-title="Hesabu" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-kg mw-list-item"><a href="https://kg.wikipedia.org/wiki/Kutanga" title="Kutanga – Kongo" lang="kg" hreflang="kg" data-title="Kutanga" data-language-autonym="Kongo" data-language-local-name="Kongo" class="interlanguage-link-target"><span>Kongo</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Haitian Creole" lang="ht" hreflang="ht" data-title="Aritmetik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Aritm%C3%A9tik" title="Aritmétik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Aritmétik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Kyrgyz" lang="ky" hreflang="ky" data-title="Арифметика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lad mw-list-item"><a href="https://lad.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Ladino" lang="lad" hreflang="lad" data-title="Aritmetika" data-language-autonym="Ladino" data-language-local-name="Ladino" class="interlanguage-link-target"><span>Ladino</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Arithmetica" title="Arithmetica – Latin" lang="la" hreflang="la" data-title="Arithmetica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Aritm%C4%93tika" title="Aritmētika – Latvian" lang="lv" hreflang="lv" data-title="Aritmētika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Lithuanian" lang="lt" hreflang="lt" data-title="Aritmetika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nia mw-list-item"><a href="https://nia.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Nias" lang="nia" hreflang="nia" data-title="Aritmetika" data-language-autonym="Li Niha" data-language-local-name="Nias" class="interlanguage-link-target"><span>Li Niha</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Aritmetica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/sapme%27ocmaci" title="sapme'ocmaci – Lojban" lang="jbo" hreflang="jbo" data-title="sapme'ocmaci" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Aritmetega" title="Aritmetega – Lombard" lang="lmo" hreflang="lmo" data-title="Aritmetega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%A1mtan" title="Számtan – Hungarian" lang="hu" hreflang="hu" data-title="Számtan" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Аритметика – Macedonian" lang="mk" hreflang="mk" data-title="Аритметика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Malagasy" lang="mg" hreflang="mg" data-title="Aritmetika" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="അങ്കഗണിതം – Malayalam" lang="ml" hreflang="ml" data-title="അങ്കഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अंकगणित – Marathi" lang="mr" hreflang="mr" data-title="अंकगणित" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%94%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="არითმეტიკა – Mingrelian" lang="xmf" hreflang="xmf" data-title="არითმეტიკა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Malay" lang="ms" hreflang="ms" data-title="Aritmetik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Mirandese" lang="mwl" hreflang="mwl" data-title="Aritmética" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA" title="Арифметик – Mongolian" lang="mn" hreflang="mn" data-title="Арифметик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%8F%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ဂဏန်းသင်္ချာ – Burmese" lang="my" hreflang="my" data-title="ဂဏန်းသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nah mw-list-item"><a href="https://nah.wikipedia.org/wiki/Tlap%C5%8Dhuall%C5%8Dtl" title="Tlapōhuallōtl – Nahuatl" lang="nah" hreflang="nah" data-title="Tlapōhuallōtl" data-language-autonym="Nāhuatl" data-language-local-name="Nahuatl" class="interlanguage-link-target"><span>Nāhuatl</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Maqusa" title="Maqusa – Fijian" lang="fj" hreflang="fj" data-title="Maqusa" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rekenen" title="Rekenen – Dutch" lang="nl" hreflang="nl" data-title="Rekenen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%99%E0%A5%8D%E0%A4%95_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अङ्क गणित – Nepali" lang="ne" hreflang="ne" data-title="अङ्क गणित" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="अंकगणित – Newari" lang="new" hreflang="new" data-title="अंकगणित" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AE%97%E8%A1%93" title="算術 – Japanese" lang="ja" hreflang="ja" data-title="算術" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Artemeteca" title="Artemeteca – Neapolitan" lang="nap" hreflang="nap" data-title="Artemeteca" data-language-autonym="Napulitano" data-language-local-name="Neapolitan" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Aritmeetik" title="Aritmeetik – Northern Frisian" lang="frr" hreflang="frr" data-title="Aritmeetik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Aritmetikk" title="Aritmetikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Aritmetikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Aritmetikk" title="Aritmetikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Aritmetikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Aritmetike" title="Aritmetike – Novial" lang="nov" hreflang="nov" data-title="Aritmetike" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Occitan" lang="oc" hreflang="oc" data-title="Aritmetica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakxii" title="Lakxii – Oromo" lang="om" hreflang="om" data-title="Lakxii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Arifmetika" title="Arifmetika – Uzbek" lang="uz" hreflang="uz" data-title="Arifmetika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A9%B0%E0%A8%95_%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਅੰਕ ਗਣਿਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਅੰਕ ਗਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8" title="حساب – Western Punjabi" lang="pnb" hreflang="pnb" data-title="حساب" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pap mw-list-item"><a href="https://pap.wikipedia.org/wiki/Aritm%C3%A9tika" title="Aritmétika – Papiamento" lang="pap" hreflang="pap" data-title="Aritmétika" data-language-autonym="Papiamentu" data-language-local-name="Papiamento" class="interlanguage-link-target"><span>Papiamentu</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8" title="حساب – Pashto" lang="ps" hreflang="ps" data-title="حساب" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Aritmitik" title="Aritmitik – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Aritmitik" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Piedmontese" lang="pms" hreflang="pms" data-title="Aritmética" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Arithmetik" title="Arithmetik – Low German" lang="nds" hreflang="nds" data-title="Arithmetik" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Arytmetyka" title="Arytmetyka – Polish" lang="pl" hreflang="pl" data-title="Arytmetyka" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Aritm%C3%A9tica" title="Aritmética – Portuguese" lang="pt" hreflang="pt" data-title="Aritmética" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Arifmetika" title="Arifmetika – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Arifmetika" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Esap" title="Esap – Crimean Tatar" lang="crh" hreflang="crh" data-title="Esap" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Aritmetic%C4%83" title="Aritmetică – Romanian" lang="ro" hreflang="ro" data-title="Aritmetică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupa_hap%27ichiy" title="Yupa hap'ichiy – Quechua" lang="qu" hreflang="qu" data-title="Yupa hap'ichiy" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%90%D1%80%D1%96%D1%84%D0%BC%D0%B5%D1%82%D1%96%D0%BA%D0%B0" title="Аріфметіка – Rusyn" lang="rue" hreflang="rue" data-title="Аріфметіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Russian" lang="ru" hreflang="ru" data-title="Арифметика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Yakut" lang="sah" hreflang="sah" data-title="Арифметика" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%85%E0%A4%99%E0%A5%8D%E0%A4%95%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D" title="अङ्कगणितम् – Sanskrit" lang="sa" hreflang="sa" data-title="अङ्कगणितम्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Aritm%C3%A8tica" title="Aritmètica – Sardinian" lang="sc" hreflang="sc" data-title="Aritmètica" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Arithmetic" title="Arithmetic – Scots" lang="sco" hreflang="sco" data-title="Arithmetic" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Albanian" lang="sq" hreflang="sq" data-title="Aritmetika" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Aritmetica" title="Aritmetica – Sicilian" lang="scn" hreflang="scn" data-title="Aritmetica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Arithmetic" title="Arithmetic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Arithmetic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D8%B1%D9%BF%D9%85%D9%8A%D9%BD%DA%AA" title="ارٿميٽڪ – Sindhi" lang="sd" hreflang="sd" data-title="ارٿميٽڪ" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Slovak" lang="sk" hreflang="sk" data-title="Aritmetika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Slovenian" lang="sl" hreflang="sl" data-title="Aritmetika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%DB%8E%D8%B1%DB%95" title="ژمێرە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمێرە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%82%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Аритметика – Serbian" lang="sr" hreflang="sr" data-title="Аритметика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Aritmetika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aritmetiikka" title="Aritmetiikka – Finnish" lang="fi" hreflang="fi" data-title="Aritmetiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Swedish" lang="sv" hreflang="sv" data-title="Aritmetik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Aritmetika" title="Aritmetika – Tagalog" lang="tl" hreflang="tl" data-title="Aritmetika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="எண்கணிதம் – Tamil" lang="ta" hreflang="ta" data-title="எண்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Tatar" lang="tt" hreflang="tt" data-title="Арифметика" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="เลขคณิต – Thai" lang="th" hreflang="th" data-title="เลขคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%B2%D0%B8%D1%81%D0%BE%D0%B1" title="Ҳисоб – Tajik" lang="tg" hreflang="tg" data-title="Ҳисоб" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Turkish" lang="tr" hreflang="tr" data-title="Aritmetik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Арифметика – Ukrainian" lang="uk" hreflang="uk" data-title="Арифметика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8" title="حساب – Urdu" lang="ur" hreflang="ur" data-title="حساب" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%A6%D8%A7%D8%B1%D9%89%D9%81%D9%85%DB%90%D8%AA%D9%89%D9%83%D8%A7" title="ئارىفمېتىكا – Uyghur" lang="ug" hreflang="ug" data-title="ئارىفمېتىكا" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Aritm%C3%A8tega" title="Aritmètega – Venetian" lang="vec" hreflang="vec" data-title="Aritmètega" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Arifmetik" title="Arifmetik – Veps" lang="vep" hreflang="vep" data-title="Arifmetik" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_h%E1%BB%8Dc" title="Số học – Vietnamese" lang="vi" hreflang="vi" data-title="Số học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Kalkulav" title="Kalkulav – Volapük" lang="vo" hreflang="vo" data-title="Kalkulav" data-language-autonym="Volapük" data-language-local-name="Volapük" class="interlanguage-link-target"><span>Volapük</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Arvokunst" title="Arvokunst – Võro" lang="vro" hreflang="vro" data-title="Arvokunst" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%AE%97%E8%A1%93" title="算術 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="算術" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Aritmetik%C3%A1" title="Aritmetiká – Waray" lang="war" hreflang="war" data-title="Aritmetiká" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%AE%97%E6%9C%AF" title="算术 – Wu" lang="wuu" hreflang="wuu" data-title="算术" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Tinhlayo-tinomboro" title="Tinhlayo-tinomboro – Tsonga" lang="ts" hreflang="ts" data-title="Tinhlayo-tinomboro" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target"><span>Xitsonga</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%99%D7%98%D7%9E%D7%A2%D7%98%D7%99%D7%A7" title="אריטמעטיק – Yiddish" lang="yi" hreflang="yi" data-title="אריטמעטיק" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8C%E1%B9%A3%C3%ADr%C3%B2" title="Ìṣírò – Yoruba" lang="yo" hreflang="yo" data-title="Ìṣírò" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%AE%97%E8%A1%93" title="算術 – Cantonese" lang="yue" hreflang="yue" data-title="算術" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Rekenen" title="Rekenen – Zeelandic" lang="zea" hreflang="zea" data-title="Rekenen" data-language-autonym="Zeêuws" data-language-local-name="Zeelandic" class="interlanguage-link-target"><span>Zeêuws</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Ar%C4%97tmet%C4%97ka" title="Arėtmetėka – Samogitian" lang="sgs" hreflang="sgs" data-title="Arėtmetėka" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AE%97%E6%9C%AF" title="算术 – Chinese" lang="zh" hreflang="zh" data-title="算术" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Aritmetik" title="Aritmetik – Iban" lang="iba" hreflang="iba" data-title="Aritmetik" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-tly mw-list-item"><a href="https://tly.wikipedia.org/wiki/Hisob" title="Hisob – Talysh" lang="tly" hreflang="tly" data-title="Hisob" data-language-autonym="Tolışi" data-language-local-name="Talysh" class="interlanguage-link-target"><span>Tolışi</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11205#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Arithmetic" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Arithmetic" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Arithmetic"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Arithmetic&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Arithmetic&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Arithmetic"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Arithmetic&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Arithmetic&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Arithmetic" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Arithmetic" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Arithmetic&oldid=1272376224" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Arithmetic&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Arithmetic&id=1272376224&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FArithmetic"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FArithmetic"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Arithmetic&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Arithmetic&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Arithmetic" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Arithmetic" hreflang="en"><span>Wikibooks</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://en.wikiquote.org/wiki/Arithmetic" hreflang="en"><span>Wikiquote</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11205" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-good-star" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of elementary mathematics</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the 2004 song by Brooke Fraser, see <a href="/wiki/Arithmetic_(song)" title="Arithmetic (song)">Arithmetic (song)</a>. For the 1703 Russian textbook, see <a href="/wiki/Arithmetic_(book)" title="Arithmetic (book)"><i>Arithmetic</i> (book)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Arithmetic_operations.svg" class="mw-file-description"><img alt="Diagram of symbols of arithmetic operations" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/220px-Arithmetic_operations.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/330px-Arithmetic_operations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/440px-Arithmetic_operations.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The main arithmetic operations are addition, subtraction, multiplication, and division.</figcaption></figure> <p><b>Arithmetic</b> is an elementary branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that studies numerical operations like <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>. In a wider sense, it also includes <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, extraction of <a href="/wiki/Nth_root" title="Nth root">roots</a>, and taking <a href="/wiki/Logarithms" class="mw-redirect" title="Logarithms">logarithms</a>. </p><p>Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative <a href="/wiki/Integer" title="Integer">integers</a>. Rational number arithmetic involves operations on <a href="/wiki/Fraction" title="Fraction">fractions</a> of integers. Real number arithmetic is about calculations with <a href="/wiki/Real_number" title="Real number">real numbers</a>, which include both <a href="/wiki/Rational_number" title="Rational number">rational</a> and <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>. </p><p>Another distinction is based on the <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> employed to perform calculations. <a href="/wiki/Decimal" title="Decimal">Decimal</a> arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express <a href="/wiki/Number" title="Number">numbers</a>. <a href="/wiki/Binary_number" title="Binary number">Binary</a> arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. <a href="/wiki/Computer_arithmetic" title="Computer arithmetic">Computer arithmetic</a> deals with the specificities of the implementation of binary arithmetic on <a href="/wiki/Computer" title="Computer">computers</a>. Some arithmetic systems operate on <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> other than numbers, such as <a href="/wiki/Interval_arithmetic" title="Interval arithmetic">interval arithmetic</a> and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> arithmetic. </p><p>Arithmetic operations form the basis of many branches of mathematics, such as <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Calculus" title="Calculus">calculus</a>, and <a href="/wiki/Statistics" title="Statistics">statistics</a>. They play a similar role in the <a href="/wiki/Sciences" class="mw-redirect" title="Sciences">sciences</a>, like <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Economics" title="Economics">economics</a>. Arithmetic is present in many aspects of <a href="/wiki/Daily_life" class="mw-redirect" title="Daily life">daily life</a>, for example, to calculate change while shopping or to manage <a href="/wiki/Personal_finances" class="mw-redirect" title="Personal finances">personal finances</a>. It is one of the earliest forms of <a href="/wiki/Mathematics_education" title="Mathematics education">mathematics education</a> that students encounter. Its cognitive and conceptual foundations are studied by <a href="/wiki/Psychology" title="Psychology">psychology</a> and <a href="/wiki/Philosophy" title="Philosophy">philosophy</a>. </p><p>The practice of arithmetic is at least thousands and possibly tens of thousands of years old. <a href="/wiki/Ancient_civilizations" class="mw-redirect" title="Ancient civilizations">Ancient civilizations</a> like the <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptians</a> and the <a href="/wiki/Sumer" title="Sumer">Sumerians</a> invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the <a href="/wiki/Ancient_Greeks" class="mw-redirect" title="Ancient Greeks">ancient Greeks</a> initiated a more abstract study of numbers and introduced the method of rigorous <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proofs</a>. The <a href="/wiki/Ancient_India" class="mw-redirect" title="Ancient India">ancient Indians</a> developed the concept of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> and the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">decimal system</a>, which Arab mathematicians further refined and spread to the Western world during the medieval period. The first <a href="/wiki/Mechanical_calculator" title="Mechanical calculator">mechanical calculators</a> were invented in the 17th century. The 18th and 19th centuries saw the development of modern <a href="/wiki/Number_theory" title="Number theory">number theory</a> and the formulation of <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic foundations</a> of arithmetic. In the 20th century, the emergence of <a href="/wiki/Electronic_calculator" class="mw-redirect" title="Electronic calculator">electronic calculators</a> and computers revolutionized the accuracy and speed with which arithmetic calculations could be performed. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition,_etymology,_and_related_fields"><span id="Definition.2C_etymology.2C_and_related_fields"></span>Definition, etymology, and related fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=1" title="Edit section: Definition, etymology, and related fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Arithmetic is the fundamental branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In a wider sense, it also includes <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, extraction of <a href="/wiki/Nth_root" title="Nth root">roots</a>, and <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The term <i>arithmetic</i> has its root in the Latin term <i lang="la"><a href="https://en.wiktionary.org/wiki/arithmetica#Latin" class="extiw" title="wikt:arithmetica">arithmetica</a></i> which derives from the Ancient Greek words <span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%80%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82#Ancient_Greek" class="extiw" title="wikt:ἀριθμός">ἀριθμός</a></span> (<i>arithmos</i>), meaning <span class="gloss-quot">'</span><span class="gloss-text">number</span><span class="gloss-quot">'</span>, and <span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%80%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%B9%CE%BA%CE%AE#Ancient_Greek" class="extiw" title="wikt:ἀριθμητική">ἀριθμητική</a></span> <span lang="grc"><a href="https://en.wiktionary.org/wiki/%CF%84%CE%AD%CF%87%CE%BD%CE%B7#Ancient_Greek" class="extiw" title="wikt:τέχνη">τέχνη</a></span> (<i>arithmetike tekhne</i>), meaning <span class="gloss-quot">'</span><span class="gloss-text">the art of counting</span><span class="gloss-quot">'</span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> However, the more common view is to include operations on <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>, <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>, <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, and sometimes also <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> in its scope.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Some definitions restrict arithmetic to the field of numerical calculations.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> When understood in a wider sense, it also includes the study of how the concept of <a href="/wiki/Numbers" class="mw-redirect" title="Numbers">numbers</a> developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Arithmetic is closely related to <a href="/wiki/Number_theory" title="Number theory">number theory</a> and some authors use the terms as synonyms.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>, <a href="/wiki/Factorization" title="Factorization">factorization</a>, and <a href="/wiki/Primality" class="mw-redirect" title="Primality">primality</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Traditionally, it is known as higher arithmetic.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Numbers">Numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=2" title="Edit section: Numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Number" title="Number">Numbers</a> are <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different <a href="/wiki/Numeral_systems" class="mw-redirect" title="Numeral systems">numeral systems</a> to represent them.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kinds">Kinds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=3" title="Edit section: Kinds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Number_line.png" class="mw-file-description"><img alt="Number line showing different types of numbers" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Number_line.png/390px-Number_line.png" decoding="async" width="390" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Number_line.png/585px-Number_line.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Number_line.png/780px-Number_line.png 2x" data-file-width="1295" data-file-height="355" /></a><figcaption>Different types of numbers on a <a href="/wiki/Number_line" title="Number line">number line</a>. Integers are black, rational numbers are blue, and irrational numbers are green.</figcaption></figure> <p>The main kinds of numbers employed in arithmetic are <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, whole numbers, <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>, <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>, and <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4,...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4156750394aa576a16632a2f546bbbb30099ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.212ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4,...\}}" /></span>. The symbol of the natural numbers is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /></span>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,3,4,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,3,4,...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775096ca4879c4659498dd25f7aeb33136541842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.409ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,3,4,...\}}" /></span> and have the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab7e98123f0def29a1cd3df96a0b7a58f4202c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}}" /></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The set of integers encompasses both positive and negative whole numbers. It has the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /></span> and can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{...,-2,-1,0,1,2,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{...,-2,-1,0,1,2,...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca4a58f0db4e5db432d9c1013a33f55b1fd27ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.161ex; height:2.843ex;" alt="{\displaystyle \{...,-2,-1,0,1,2,...\}}" /></span>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Based on how natural and whole numbers are used, they can be distinguished into <a href="/wiki/Cardinal_numerals" class="mw-redirect" title="Cardinal numerals">cardinal</a> and <a href="/wiki/Ordinal_numerals" class="mw-redirect" title="Ordinal numerals">ordinal numbers</a>. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?".<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>A number is rational if it can be represented as the <a href="/wiki/Ratio" title="Ratio">ratio</a> of two integers. For instance, the rational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" /></span> is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {281}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>281</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {281}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d3aff130fab330aabd401846578e255b7decf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.302ex; height:3.676ex;" alt="{\displaystyle {\tfrac {281}{3}}}" /></span>. The set of rational numbers includes all integers, which are <a href="/wiki/Fraction" title="Fraction">fractions</a> with a denominator of 1. The symbol of the rational numbers is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /></span>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Decimal_fraction" class="mw-redirect" title="Decimal fraction">Decimal fractions</a> like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8ce81768711c9640f43853bb6d5202d42d1114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{10}}}" /></span>, and 25.12 is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2512}{100}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2512</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2512}{100}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75f1850d14641e7d2a1e5e39cf062e9b9b0ac79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.124ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2512}{100}}}" /></span>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Every rational number corresponds to a finite or a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Square_root_of_2_triangle.svg" class="mw-file-description"><img alt="Diagram of a right triangle" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/220px-Square_root_of_2_triangle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/330px-Square_root_of_2_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/440px-Square_root_of_2_triangle.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Irrational numbers are sometimes required to describe magnitudes in <a href="/wiki/Geometry" title="Geometry">geometry</a>. For example, the length of the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> is irrational if its legs have a length of 1.</figcaption></figure> <p><a href="/wiki/Irrational_numbers" class="mw-redirect" title="Irrational numbers">Irrational numbers</a> are numbers that cannot be expressed through the ratio of two integers. They are often required to describe geometric magnitudes. For example, if a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> has legs of the length 1 then the length of its <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> is given by the irrational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>. <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> is another irrational number and describes the ratio of a <a href="/wiki/Circle" title="Circle">circle</a>'s <a href="/wiki/Circumference" title="Circumference">circumference</a> to its <a href="/wiki/Diameter" title="Diameter">diameter</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The decimal representation of an irrational number is infinite without repeating decimals.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Even wider classes of numbers include <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> and <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Numeral_systems">Numeral systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=4" title="Edit section: Numeral systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a></div> <p>A <a href="/wiki/Numerical_digit" title="Numerical digit">numeral</a> is a symbol to represent a number and numeral systems are representational frameworks.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Numeral systems are either <a href="/wiki/Positional" class="mw-redirect" title="Positional">positional</a> or non-positional. All early numeral systems were non-positional.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> For non-positional numeral systems, the value of a digit does not depend on its position in the numeral.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:242px;max-width:242px"><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:47px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tally_marks.svg" class="mw-file-description"><img alt="Diagram showing tally marks" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tally_marks.svg/238px-Tally_marks.svg.png" decoding="async" width="238" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tally_marks.svg/357px-Tally_marks.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tally_marks.svg/476px-Tally_marks.svg.png 2x" data-file-width="512" data-file-height="102" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:149px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Vl%C4%8D%C3%AD_radius.jpg" class="mw-file-description"><img alt="Photo of tally sticks" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Vl%C4%8D%C3%AD_radius.jpg/238px-Vl%C4%8D%C3%AD_radius.jpg" decoding="async" width="238" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Vl%C4%8D%C3%AD_radius.jpg/357px-Vl%C4%8D%C3%AD_radius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Vl%C4%8D%C3%AD_radius.jpg/476px-Vl%C4%8D%C3%AD_radius.jpg 2x" data-file-width="800" data-file-height="502" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a> and some <a href="/wiki/Tally_sticks" class="mw-redirect" title="Tally sticks">tally sticks</a> use the non-positional <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a>.</div></div></div></div> <p>The simplest non-positional system is the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a>. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Variations of the unary numeral systems are employed in <a href="/wiki/Tally_stick" title="Tally stick">tally sticks</a> using dents and in <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a>.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Hieroglyph_numerals.svg" class="mw-file-description"><img alt="Diagram of hieroglyphic numerals" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Hieroglyph_numerals.svg/220px-Hieroglyph_numerals.svg.png" decoding="async" width="220" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Hieroglyph_numerals.svg/330px-Hieroglyph_numerals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Hieroglyph_numerals.svg/440px-Hieroglyph_numerals.svg.png 2x" data-file-width="512" data-file-height="171" /></a><figcaption>Hieroglyphic numerals from 1 to 10,000<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><a href="/wiki/Egyptian_hieroglyphics" class="mw-redirect" title="Egyptian hieroglyphics">Egyptian hieroglyphics</a> had a more complex non-positional <a href="/wiki/Egyptian_numerals" title="Egyptian numerals">numeral system</a>. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the <a href="/wiki/Roman_numeral_system" class="mw-redirect" title="Roman numeral system">Roman numeral system</a>. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a <a href="/wiki/Radix" title="Radix">radix</a> that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>, the radix is 10. This means that the first digit is multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d6ef004bb97df6bf6ee2ec20f8eec7aa32adcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{0}}" /></span>, the next digit is multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f89eb50e31c21a232c81e0c880681945a550fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{1}}" /></span>, and so on. For example, the decimal numeral 532 stands for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4925712e4410bbd891c20d51559ea82c50270d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.343ex; height:2.843ex;" alt="{\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}}" /></span>. Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Another positional numeral system used extensively in <a href="/wiki/Computer_arithmetic" title="Computer arithmetic">computer arithmetic</a> is the <a href="/wiki/Binary_system" title="Binary system">binary system</a>, which has a radix of 2. This means that the first digit is multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75edddb13f7181972ba01302b2eb0d09ebcf24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{0}}" /></span>, the next digit by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f990b8febae3ab32d873486ee5c343e8db92ceca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{1}}" /></span>, and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3d29b81f0d05c111da842c044a1cfdff3bccff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.754ex; height:2.843ex;" alt="{\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}" /></span>. In computing, each digit in the binary notation corresponds to one <a href="/wiki/Bit" title="Bit">bit</a>.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The earliest positional system was developed by <a href="/wiki/Ancient_Babylonians" class="mw-redirect" title="Ancient Babylonians">ancient Babylonians</a> and had a radix of 60.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Operations">Operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=5" title="Edit section: Operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:55px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Apple_addition.svg" class="mw-file-description"><img alt="Diagram of addition" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Apple_addition.svg/288px-Apple_addition.svg.png" decoding="async" width="288" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Apple_addition.svg/432px-Apple_addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Apple_addition.svg/576px-Apple_addition.svg.png 2x" data-file-width="512" data-file-height="99" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:90px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Apple_division.svg" class="mw-file-description"><img alt="Diagram of division" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Apple_division.svg/288px-Apple_division.svg.png" decoding="async" width="288" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Apple_division.svg/432px-Apple_division.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Apple_division.svg/576px-Apple_division.svg.png 2x" data-file-width="512" data-file-height="161" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Arithmetic operations underlie many everyday occurrences, like when putting four apples from one bag together with three apples from another bag (top image) or when distributing nine apples equally among three children (bottom image).</div></div></div></div> <p>Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> that have numbers both as input and output.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> The most important operations in arithmetic are <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Further operations include <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, extraction of <a href="/wiki/Nth_root" title="Nth root">roots</a>, and <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> If these operations are performed on variables rather than numbers, they are sometimes referred to as <a href="/wiki/Algebraic_operations" class="mw-redirect" title="Algebraic operations">algebraic operations</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p><p>Two important concepts in relation to arithmetic operations are <a href="/wiki/Identity_element" title="Identity element">identity elements</a> and <a href="/wiki/Inverse_element" title="Inverse element">inverse elements</a>. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> of the number 6 is -6 since their sum is 0.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>There are not only inverse elements but also <a href="/wiki/Inverse_function" title="Inverse function">inverse operations</a>. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13+4-4=13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13+4-4=13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339bef3b8c7ad200f16fda73e828200afd425575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.754ex; height:2.343ex;" alt="{\displaystyle 13+4-4=13}" /></span>. Defined more formally, the operation "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }" /></span>" is an inverse of the operation "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" /></span>" if it fulfills the following condition: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\star s=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>⋆</mo> <mi>s</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\star s=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea353417709dd47077cf91222b62f43be1d7b7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.272ex; height:2.009ex;" alt="{\displaystyle t\star s=r}" /></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\circ s=t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∘</mo> <mi>s</mi> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\circ s=t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711c2f730193a3da009aa15a6f4ca83a02e7ce2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.272ex; height:2.009ex;" alt="{\displaystyle r\circ s=t}" /></span>.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> and <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7+9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>+</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7+9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147b7081a0c3192e3f7ea6d795275cc4e2150d06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 7+9}" /></span> is the same as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9+7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732742e19719a19749bc56f9692e91f0e06420d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 9+7}" /></span>. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (5\times 4)\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>5</mn> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (5\times 4)\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ceab64844d991621a526a74349541092d2c501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.977ex; height:2.843ex;" alt="{\displaystyle (5\times 4)\times 2}" /></span> is the same as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\times (4\times 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>×</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\times (4\times 2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724325d087b83a4c4743ece49ede2ee7afcf43af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.977ex; height:2.843ex;" alt="{\displaystyle 5\times (4\times 2)}" /></span>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Addition_and_subtraction">Addition and subtraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=6" title="Edit section: Addition and subtraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Addition" title="Addition">Addition</a> and <a href="/wiki/Subtraction" title="Subtraction">Subtraction</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:75px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Addition1.png" class="mw-file-description"><img alt="Diagram showing addition" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Addition1.png/288px-Addition1.png" decoding="async" width="288" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Addition1.png/432px-Addition1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Addition1.png/576px-Addition1.png 2x" data-file-width="977" data-file-height="255" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:68px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Subtraction.png" class="mw-file-description"><img alt="Diagram showing subtraction" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Subtraction.png/288px-Subtraction.png" decoding="async" width="288" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Subtraction.png/432px-Subtraction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Subtraction.png/576px-Subtraction.png 2x" data-file-width="1129" data-file-height="269" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Addition and subtraction</div></div></div></div> <p>Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span>. Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+2=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+2=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6a3bbd9401d92584246527beee883714954a41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2+2=4}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6.3+1.26=7.56}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6.3</mn> <mo>+</mo> <mn>1.26</mn> <mo>=</mo> <mn>7.56</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6.3+1.26=7.56}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16724ee7008c87bf45fcda8638a036eed9f9b0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.179ex; height:2.343ex;" alt="{\displaystyle 6.3+1.26=7.56}" /></span>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> The term <a href="/wiki/Summation" title="Summation">summation</a> is used if several additions are performed in a row.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Counting is a type of repeated addition in which the number 1 is continuously added.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" /></span>.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14-8=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>14</mn> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14-8=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140fcf0a40277c13561d1cc37e387046b532703e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.589ex; height:2.343ex;" alt="{\displaystyle 14-8=6}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45-1.7=43.3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>45</mn> <mo>−</mo> <mn>1.7</mn> <mo>=</mo> <mn>43.3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45-1.7=43.3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c16b78fd7c53595a8eabc080b1c4d7cc9d5f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.37ex; height:2.343ex;" alt="{\displaystyle 45-1.7=43.3}" /></span>. Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14-8=14+(-8)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>14</mn> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mn>14</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14-8=14+(-8)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e773728e1838db8a44318e0ee2eb8140f12637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.371ex; height:2.843ex;" alt="{\displaystyle 14-8=14+(-8)}" /></span>. This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>The additive identity element is 0 and the additive inverse of a number is the negative of that number. For instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13+0=13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13+0=13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a853c7ee65f44968ff1f67eb1ba7e6e3dc6433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.751ex; height:2.343ex;" alt="{\displaystyle 13+0=13}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13+(-13)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>13</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13+(-13)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940882c06ac36da3ed61c6d77fd2eae6660a02f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.369ex; height:2.843ex;" alt="{\displaystyle 13+(-13)=0}" /></span>. Addition is both commutative and associative.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiplication_and_division">Multiplication and division</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=7" title="Edit section: Multiplication and division"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division (mathematics)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:71px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Multiplication1.png" class="mw-file-description"><img alt="Diagram showing multiplication" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Multiplication1.png/288px-Multiplication1.png" decoding="async" width="288" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Multiplication1.png/432px-Multiplication1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Multiplication1.png/576px-Multiplication1.png 2x" data-file-width="1038" data-file-height="258" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:68px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Division1.png" class="mw-file-description"><img alt="Diagram showing division" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Division1.png/288px-Division1.png" decoding="async" width="288" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Division1.png/432px-Division1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Division1.png/576px-Division1.png 2x" data-file-width="1103" data-file-height="263" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Multiplication and division</div></div></div></div> <p>Multiplication is an arithmetic operation in which two numbers, called the multiplier and the multiplicand, are combined into a single number called the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> The symbols of multiplication are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" /></span>, and *. Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 3=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b249223f0c54e210f6ceeea7ee975b852460448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.426ex; height:2.176ex;" alt="{\displaystyle 2\times 3=6}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.3\cdot 5=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.3</mn> <mo>⋅</mo> <mn>5</mn> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.3\cdot 5=1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d70d0edef2375992f22c53c41092dd48a3436b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.884ex; height:2.176ex;" alt="{\displaystyle 0.3\cdot 5=1.5}" /></span>. If the multiplicand is a natural number then multiplication is the same as repeated addition, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 3=2+2+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3=2+2+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971fe3575551d98473ed782e872150d008c4c01a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.432ex; height:2.343ex;" alt="{\displaystyle 2\times 3=2+2+2}" /></span>.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the <a href="/wiki/Quotient" title="Quotient">quotient</a>. The symbols of division are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle /}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle /}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0c4de1fba637d9799f6c64a6c77bf016d0ce1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle /}" /></span>. Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 48\div 8=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>48</mn> <mo>÷</mo> <mn>8</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 48\div 8=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7522e3bcbb5970db2d3e56ad151c1bd0432bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.589ex; height:2.176ex;" alt="{\displaystyle 48\div 8=6}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 29.4/1.4=21}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>29.4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1.4</mn> <mo>=</mo> <mn>21</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 29.4/1.4=21}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0badc5ac18eac9dbeddfa2f4a7fe470b61c409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.692ex; height:2.843ex;" alt="{\displaystyle 29.4/1.4=21}" /></span>.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a>. The reciprocal of a number is 1 divided by that number. For instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 48\div 8=48\times {\tfrac {1}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>48</mn> <mo>÷</mo> <mn>8</mn> <mo>=</mo> <mn>48</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 48\div 8=48\times {\tfrac {1}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e810140c4c5253fa12ba889457ef304f060c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.25ex; height:3.676ex;" alt="{\displaystyle 48\div 8=48\times {\tfrac {1}{8}}}" /></span>.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13\times 1=13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>×</mo> <mn>1</mn> <mo>=</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13\times 1=13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927f3b6bf371f7fbbca802f41df57cf2522ef409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.751ex; height:2.176ex;" alt="{\displaystyle 13\times 1=13}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13\times {\tfrac {1}{13}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>13</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13\times {\tfrac {1}{13}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcc6598916570ce467d1c409427bbe7f5e4fe19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.906ex; height:3.676ex;" alt="{\displaystyle 13\times {\tfrac {1}{13}}=1}" /></span>. Multiplication is both commutative and associative.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Exponentiation_and_logarithm">Exponentiation and logarithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=8" title="Edit section: Exponentiation and logarithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> and <a href="/wiki/Logarithm" title="Logarithm">Logarithm</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:168px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Exponentiation.png" class="mw-file-description"><img alt="Diagram showing exponentiation" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exponentiation.png/288px-Exponentiation.png" decoding="async" width="288" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exponentiation.png/432px-Exponentiation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exponentiation.png/576px-Exponentiation.png 2x" data-file-width="606" data-file-height="355" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:105px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Logarithm1.png" class="mw-file-description"><img alt="Diagram showing logarithm" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Logarithm1.png/288px-Logarithm1.png" decoding="async" width="288" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Logarithm1.png/432px-Logarithm1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Logarithm1.png/576px-Logarithm1.png 2x" data-file-width="1029" data-file-height="378" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Exponentiation and logarithm</div></div></div></div> <p>Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in <a href="/wiki/Superscript" class="mw-redirect" title="Superscript">superscript</a> right after the base. Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497cd52a6b919afe9294de98282cccc6c74b5fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.676ex;" alt="{\displaystyle 2^{4}=16}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" /></span>^<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3=27}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>=</mo> <mn>27</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3=27}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690cca215ddc10b4a90bd0a4c687b466843c273a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.586ex; height:2.176ex;" alt="{\displaystyle 3=27}" /></span>. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=2\times 2\times 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=2\times 2\times 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7293a7c25df8be2753bf517f91ee8703df87ae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.486ex; height:2.676ex;" alt="{\displaystyle 2^{4}=2\times 2\times 2\times 2}" /></span>.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> </p><p>Roots are a special type of exponentiation using a fractional exponent. For example, the <a href="/wiki/Square_root" title="Square root">square root</a> of a number is the same as raising the number to the power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" /></span> and the <a href="/wiki/Cube_root" title="Cube root">cube root</a> of a number is the same as raising the number to the power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efc37d09854a3f8fb997e7de4331876bc49c2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}}" /></span>. Examples are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bd54c0c8ec8d16482836ee1559c3d1196dfb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.356ex; height:3.843ex;" alt="{\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d751796d4bcce06077dfdd0751dd98c2cb988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.681ex; height:3.843ex;" alt="{\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3}" /></span>.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p><p>Logarithm is the inverse of exponentiation. The logarithm of a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> to the base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span> is the <a href="/wiki/Exponent" class="mw-redirect" title="Exponent">exponent</a> to which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span> must be raised to produce <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>. For instance, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1000=10^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1000</mn> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1000=10^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d647d0505284ca7d5f07096168e45f2f9521ec0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.127ex; height:2.676ex;" alt="{\displaystyle 1000=10^{3}}" /></span>, the logarithm base 10 of 1000 is 3. The logarithm of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> to base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cad5f91f24392d04f1c9324d7154401e1ce3bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.048ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x)}" /></span>, or without parentheses, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777b35d8d91f445f98286c9619003039e28dfe94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.626ex; height:2.676ex;" alt="{\displaystyle \log _{b}x}" /></span>, or even without the explicit base, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d453de713a8c45f7bf99108752531ed7d6dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.689ex; height:2.509ex;" alt="{\displaystyle \log x}" /></span>, when the base can be understood from context. So, the previous example can be written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}1000=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>1000</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}1000=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6c39910e86cae18a3cc803e7bf66c0be8e3d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.146ex; height:2.676ex;" alt="{\displaystyle \log _{10}1000=3}" /></span>.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 14^{1}=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>14</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14^{1}=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5234113a37619e1ec28672fc3ed81dbe88699adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.803ex; height:2.676ex;" alt="{\displaystyle 14^{1}=14}" /></span>. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> Exponentiation and logarithm are neither commutative nor associative.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=9" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Integer_arithmetic">Integer arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=10" title="Edit section: Integer arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Number_line_method.svg" class="mw-file-description"><img alt="Diagram of number line method" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Number_line_method.svg/290px-Number_line_method.svg.png" decoding="async" width="290" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Number_line_method.svg/435px-Number_line_method.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Number_line_method.svg/580px-Number_line_method.svg.png 2x" data-file-width="512" data-file-height="83" /></a><figcaption>Using the number line method, calculating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/017132389bd9a6ecd33ee13c39fe272492eba331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 5+2}" /></span> is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend.</figcaption></figure> <p>Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an <a href="/wiki/Addition_table" class="mw-redirect" title="Addition table">addition table</a> or a <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a>. Other common methods are verbal <a href="/wiki/Counting" title="Counting">counting</a> and <a href="/wiki/Finger-counting" title="Finger-counting">finger-counting</a>.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="display:inline-table; text-align:center; margin:1em .8em 1em 1.6em;"> <caption>Addition table </caption> <tbody><tr> <th>+</th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>... </th></tr> <tr> <th>0 </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>... </td></tr> <tr> <th>1 </th> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>... </td></tr> <tr> <th>2 </th> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>... </td></tr> <tr> <th>3 </th> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>... </td></tr> <tr> <th scope="row">4 </th> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>... </td></tr> <tr> <th scope="row">... </th> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>... </td></tr></tbody></table> <table class="wikitable" style="display:inline-table; text-align:center; margin: 1em .8em 1em 1.6em;"> <caption>Multiplication table </caption> <tbody><tr> <th>×</th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>... </th></tr> <tr> <th>0 </th> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>... </td></tr> <tr> <th>1 </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>... </td></tr> <tr> <th>2 </th> <td>0</td> <td>2</td> <td>4</td> <td>6</td> <td>8</td> <td>... </td></tr> <tr> <th>3 </th> <td>0</td> <td>3</td> <td>6</td> <td>9</td> <td>12</td> <td>... </td></tr> <tr> <th scope="row">4 </th> <td>0</td> <td>4</td> <td>8</td> <td>12</td> <td>16</td> <td>... </td></tr> <tr> <th scope="row">... </th> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>... </td></tr></tbody></table> <div style="clear:both;" class=""></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:189px;max-width:189px"><div class="thumbimage" style="height:190px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Addition_with_carry.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Addition_with_carry.png/187px-Addition_with_carry.png" decoding="async" width="187" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Addition_with_carry.png/281px-Addition_with_carry.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Addition_with_carry.png/374px-Addition_with_carry.png 2x" data-file-width="449" data-file-height="457" /></a></span></div><div class="thumbcaption">Example of <a href="/wiki/Carry_(arithmetic)" title="Carry (arithmetic)">addition with carry</a>. The black numbers are the addends, the green number is the carry, and the blue number is the sum.</div></div><div class="tsingle" style="width:199px;max-width:199px"><div class="thumbimage" style="height:190px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Long_multiplication.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Long_multiplication.png/197px-Long_multiplication.png" decoding="async" width="197" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Long_multiplication.png/296px-Long_multiplication.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Long_multiplication.png/394px-Long_multiplication.png 2x" data-file-width="670" data-file-height="648" /></a></span></div><div class="thumbcaption">Example of long multiplication. The black numbers are the multiplier and the multiplicand. The green numbers are intermediary products gained by multiplying the multiplier with only one digit of the multiplicand. The blue number is the total product calculated by adding the intermediary products.</div></div></div></div></div> <p>For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method <a href="/wiki/Carry_(arithmetic)" title="Carry (arithmetic)">addition with carries</a>, the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> Other methods used for integer additions are the <a href="/wiki/Number_line" title="Number line">number line</a> method, the partial sum method, and the compensation method.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p><p>A basic technique of integer multiplication employs repeated addition. For example, the product of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda443e7a6e78fa880a6dccbf8bdf43a10d9988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 4}" /></span> can be calculated as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+3+3+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+3+3+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c721ddf66dd5e35bfed0b3bf4766326d7bf0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.171ex; height:2.343ex;" alt="{\displaystyle 3+3+3+3}" /></span>.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> A common technique for multiplication with larger numbers is called <a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">long multiplication</a>. This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with the rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> Other techniques used for multiplication are the <a href="/wiki/Grid_method" class="mw-redirect" title="Grid method">grid method</a> and the <a href="/wiki/Lattice_method" class="mw-redirect" title="Lattice method">lattice method</a>.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> Computer science is interested in <a href="/wiki/Multiplication_algorithms" class="mw-redirect" title="Multiplication algorithms">multiplication algorithms</a> with a low <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a> to be able to efficiently multiply very large integers, such as the <a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba algorithm</a>, the <a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a>, and the <a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook algorithm</a>.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> A common technique used for division is called <a href="/wiki/Long_division" title="Long division">long division</a>. Other methods include <a href="/wiki/Short_division" title="Short division">short division</a> and <a href="/wiki/Chunking_(division)" title="Chunking (division)">chunking</a>.<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p><p>Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> One way to ensure that the result is an integer is to <a href="/wiki/Rounding" title="Rounding">round</a> the result to a whole number. However, this method leads to inaccuracies as the original value is altered.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> Another method is to perform the division only partially and retain the <a href="/wiki/Remainder" title="Remainder">remainder</a>. For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p><p>A simple method to calculate <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> is by repeated multiplication. For instance, the exponentiation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa095843f659070e722e32e92889e8eebcc517b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 3^{4}}" /></span> can be calculated as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3\times 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3\times 3\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a34c5b41409076d789f6a79b463389cc8ad449a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.171ex; height:2.176ex;" alt="{\displaystyle 3\times 3\times 3\times 3}" /></span>.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> A more efficient technique used for large exponents is <a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">exponentiation by squaring</a>. It breaks down the calculation into a number of squaring operations. For example, the exponentiation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{65}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>65</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{65}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f78370e58528b0d5131576126b8288381343eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.039ex; height:2.676ex;" alt="{\displaystyle 3^{65}}" /></span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd4f4abbfc71e582671a5a03b443195484c9260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.537ex; height:3.176ex;" alt="{\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3}" /></span>. By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than the 64 operations required for regular repeated multiplication.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> Methods to calculate <a href="/wiki/Logarithm" title="Logarithm">logarithms</a> include the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> and <a href="/wiki/Continued_fraction" title="Continued fraction">continued fractions</a>.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Number_theory">Number theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=11" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Number_theory" title="Number theory">Number theory</a></div> <p>Number theory studies the structure and properties of integers as well as the relations and laws between them.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> Some of the main branches of modern number theory include <a href="/wiki/Elementary_number_theory" class="mw-redirect" title="Elementary number theory">elementary number theory</a>, <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, and <a href="/wiki/Geometric_number_theory" class="mw-redirect" title="Geometric number theory">geometric number theory</a>.<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>, <a href="/wiki/Factorization" title="Factorization">factorization</a>, and <a href="/wiki/Primality" class="mw-redirect" title="Primality">primality</a>.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like <a href="/wiki/Prime_number_theorem" title="Prime number theorem">how prime numbers are distributed</a> and the claim that <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">every even number is a sum of two prime numbers</a>.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> and <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, as in <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> like the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a>. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane.<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> Further branches of number theory are <a href="/wiki/Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, which employs methods from <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>,<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Combinatorial_number_theory" class="mw-redirect" title="Combinatorial number theory">combinatorial number theory</a>, which relies on the field of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>,<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Computational_number_theory" title="Computational number theory">computational number theory</a>, which approaches number-theoretic problems with computational methods,<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> and applied number theory, which examines the application of number theory to fields like <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Biology" title="Biology">biology</a>, and <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>.<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> </p><p>Influential theorems in number theory include the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, <a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a>, and <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>.<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the <a href="/wiki/Number_18" class="mw-redirect" title="Number 18">number 18</a> is not a prime number and can be represented as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4c2675585f5cbb827dd05847bf28ac196da15d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.168ex; height:2.176ex;" alt="{\displaystyle 2\times 3\times 3}" /></span>, all of which are prime numbers. The <a href="/wiki/Number_19" class="mw-redirect" title="Number 19">number 19</a>, by contrast, is a prime number that has no other prime factorization.<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> Euclid's theorem states that there are infinitely many prime numbers.<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> Fermat's Last Theorem is the statement that no positive integer values exist for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span> that solve the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}+b^{n}=c^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}+b^{n}=c^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2e31ced64b8cef38ab186ec86755ecc47c861f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.828ex; height:2.509ex;" alt="{\displaystyle a^{n}+b^{n}=c^{n}}" /></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is greater than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" /></span>.<sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rational_number_arithmetic">Rational number arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=12" title="Edit section: Rational number arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a <a href="/wiki/Ratio" title="Ratio">ratio</a> of two integers.<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have the same denominator then they can be added by adding their numerators and keeping the common denominator. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b76f90e4b9aca0b04535656af0a55be506accf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.913ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}}" /></span>. A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find a common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98277e883171831d8f98897f1d31a49022dbbe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.982ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}}" /></span>.<sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> </p><p>Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383877edd8c24557de755cf69f45018fd4603abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.61ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}}" /></span>. Dividing one rational number by another can be achieved by multiplying the first number with the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of the second number. This means that the numerator and the denominator of the second number change position. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>21</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c4552909649bdff2ad3c617513c5d45e387f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.926ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}}" /></span>.<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> Unlike integer arithmetic, rational number arithmetic is closed under division as long as the divisor is not 0.<sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> </p><p>Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm.<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the <a href="/wiki/Nth_root" title="Nth root">nth root</a> of the result based on the denominator of the exponent. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24534a05788d79b1b344c2fa94cf09aeca40221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.537ex; height:3.843ex;" alt="{\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}}" /></span>. The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, which uses a series of steps to gradually refine an initial guess until it reaches the desired level of accuracy.<sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> The Taylor series or the continued fraction method can be utilized to calculate logarithms.<sup id="cite_ref-104" class="reference"><a href="#cite_note-104"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fraction</a> notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1de894f8f238d18b9d57b742b8dcc52eadb493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{10}}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {371}{100}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>371</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {371}{100}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79fb40d8b0672a463b0e28243611fca21fa1141f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.302ex; height:3.843ex;" alt="{\displaystyle {\tfrac {371}{100}}}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {44}{10000}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>44</mn> <mn>10000</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {44}{10000}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9be425194745e4f5f8652fdd084052b7be20ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.946ex; height:3.676ex;" alt="{\displaystyle {\tfrac {44}{10000}}}" /></span> are written as 0.1, 3.71, and 0.0044 in the decimal fraction notation.<sup id="cite_ref-105" class="reference"><a href="#cite_note-105"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup> Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions.<sup id="cite_ref-106" class="reference"><a href="#cite_note-106"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> Not all rational numbers have a finite representation in the decimal notation. For example, the rational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efc37d09854a3f8fb997e7de4331876bc49c2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}}" /></span> corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> is 0.<span style="text-decoration:overline;">3</span>.<sup id="cite_ref-107" class="reference"><a href="#cite_note-107"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup> Every repeating decimal expresses a rational number.<sup id="cite_ref-108" class="reference"><a href="#cite_note-108"><span class="cite-bracket">[</span>103<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Real_number_arithmetic">Real number arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=13" title="Edit section: Real number arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>.<sup id="cite_ref-109" class="reference"><a href="#cite_note-109"><span class="cite-bracket">[</span>104<span class="cite-bracket">]</span></a></sup> Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1.<sup id="cite_ref-110" class="reference"><a href="#cite_note-110"><span class="cite-bracket">[</span>105<span class="cite-bracket">]</span></a></sup> </p><p>Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a202f4739f09ab6b46aec28c27bb250022b613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.271ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+\pi }" /></span> or <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b582f4b974c6e9e31f3b448b42d8704a1b50f92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.861ex; height:2.843ex;" alt="{\displaystyle e\cdot {\sqrt {3}}}" /></span>.</span><sup id="cite_ref-111" class="reference"><a href="#cite_note-111"><span class="cite-bracket">[</span>106<span class="cite-bracket">]</span></a></sup> In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by <a href="/wiki/Truncation" title="Truncation">truncation</a> or <a href="/wiki/Rounding" title="Rounding">rounding</a>. For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number <span class="texhtml mvar" style="font-style:italic;">π</span> has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number <span class="texhtml mvar" style="font-style:italic;">π</span> is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to <span class="texhtml mvar" style="font-style:italic;">π</span> than 3.141.<sup id="cite_ref-112" class="reference"><a href="#cite_note-112"><span class="cite-bracket">[</span>107<span class="cite-bracket">]</span></a></sup> These methods allow computers to efficiently perform approximate calculations on real numbers.<sup id="cite_ref-113" class="reference"><a href="#cite_note-113"><span class="cite-bracket">[</span>108<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Approximations_and_errors">Approximations and errors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=14" title="Edit section: Approximations and errors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In science and engineering, numbers represent estimates of physical quantities derived from <a href="/wiki/Measurement" title="Measurement">measurement</a> or modeling. Unlike mathematically exact numbers such as <span class="texhtml mvar" style="font-style:italic;">π</span> or <span class="nowrap"><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>⁠</span>,</span> scientifically relevant numerical data are inherently inexact, involving some <a href="/wiki/Measurement_uncertainty" title="Measurement uncertainty">measurement uncertainty</a>.<sup id="cite_ref-114" class="reference"><a href="#cite_note-114"><span class="cite-bracket">[</span>109<span class="cite-bracket">]</span></a></sup> One basic way to express the degree of certainty about each number's value and avoid <a href="/wiki/False_precision" title="False precision">false precision</a> is to round each measurement to a certain number of digits, called <a href="/wiki/Significant_digit" class="mw-redirect" title="Significant digit">significant digits</a>, which are implied to be accurate. For example, a person's height measured with a <a href="/wiki/Tape_measure" title="Tape measure">tape measure</a> might only be precisely known to the nearest centimeter, so should be presented as 1.62 meters rather than 1.6217 meters. If converted to imperial units, this quantity should be rounded to 64 inches or 63.8 inches rather than 63.7795 inches, to clearly convey the precision of the measurement. When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly considered to be non-significant.<sup id="cite_ref-115" class="reference"><a href="#cite_note-115"><span class="cite-bracket">[</span>110<span class="cite-bracket">]</span></a></sup> For example, the numbers 0.056 and 1200 each have only 2 significant digits, but the number 40.00 has 4 significant digits. Representing uncertainty using only significant digits is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the <a href="/wiki/Approximation_error" title="Approximation error">approximation error</a> is a more sophisticated approach.<sup id="cite_ref-116" class="reference"><a href="#cite_note-116"><span class="cite-bracket">[</span>111<span class="cite-bracket">]</span></a></sup> In the example, the person's height might be represented as <span class="nowrap">1.62 ± 0.005</span> meters or <span class="nowrap">63.8 ± 0.2 inches</span>.<sup id="cite_ref-117" class="reference"><a href="#cite_note-117"><span class="cite-bracket">[</span>112<span class="cite-bracket">]</span></a></sup> </p><p>In performing calculations with uncertain quantities, the <a href="/wiki/Propagation_of_uncertainty" title="Propagation of uncertainty">uncertainty should be propagated</a> to calculated quantities. When adding or subtracting two or more quantities, add the <a href="/wiki/Absolute_uncertainty" class="mw-redirect" title="Absolute uncertainty">absolute uncertainties</a> of each summand together to obtain the absolute uncertainty of the sum. When multiplying or dividing two or more quantities, add the <a href="/wiki/Relative_uncertainty" class="mw-redirect" title="Relative uncertainty">relative uncertainties</a> of each factor together to obtain the relative uncertainty of the product.<sup id="cite_ref-118" class="reference"><a href="#cite_note-118"><span class="cite-bracket">[</span>113<span class="cite-bracket">]</span></a></sup> When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to the leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors.<sup id="cite_ref-119" class="reference"><a href="#cite_note-119"><span class="cite-bracket">[</span>114<span class="cite-bracket">]</span></a></sup> (See <a href="/wiki/Significant_figures#Arithmetic" title="Significant figures">Significant figures § Arithmetic</a>.) </p><p>More sophisticated methods of dealing with uncertain values include <a href="/wiki/Interval_arithmetic" title="Interval arithmetic">interval arithmetic</a> and <a href="/wiki/Affine_arithmetic" title="Affine arithmetic">affine arithmetic</a>. Interval arithmetic describes operations on <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>. Intervals can be used to represent a range of values if one does not know the precise magnitude, for example, because of <a href="/wiki/Measurement_error" class="mw-redirect" title="Measurement error">measurement errors</a>. Interval arithmetic includes operations like addition and multiplication on intervals, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,2]+[3,4]=[4,6]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2]+[3,4]=[4,6]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c2532e4d462b1559c78a8874474f529699957d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.896ex; height:2.843ex;" alt="{\displaystyle [1,2]+[3,4]=[4,6]}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,2]\times [3,4]=[3,8]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2]\times [3,4]=[3,8]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0145d52e8a9825828238beb80115ed39c5bb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.896ex; height:2.843ex;" alt="{\displaystyle [1,2]\times [3,4]=[3,8]}" /></span>.<sup id="cite_ref-120" class="reference"><a href="#cite_note-120"><span class="cite-bracket">[</span>115<span class="cite-bracket">]</span></a></sup> It is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude.<sup id="cite_ref-121" class="reference"><a href="#cite_note-121"><span class="cite-bracket">[</span>116<span class="cite-bracket">]</span></a></sup> </p><p>The precision of numerical quantities can be expressed uniformly using <a href="/wiki/Normalized_number" title="Normalized number">normalized scientific notation</a>, which is also convenient for concisely representing numbers which are much larger or smaller than 1. Using scientific notation, a number is decomposed into the product of a number between 1 and 10, called the <i><a href="/wiki/Significand" title="Significand">significand</a></i>, and 10 raised to some integer power, called the <i>exponent</i>. The significand consists of the significant digits of the number, and is written as a leading digit 1–9 followed by a decimal point and a sequence of digits 0–9. For example, the normalized scientific notation of the number 8276000 is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8.276\times 10^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8.276</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8.276\times 10^{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae090f50179e329d8e52034ea0a9f00455707e64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.516ex; height:2.676ex;" alt="{\displaystyle 8.276\times 10^{6}}" /></span> with significand 8.276 and exponent 6, and the normalized scientific notation of the number 0.00735 is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7.35\times 10^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7.35</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7.35\times 10^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62f2184bb6ddef7648b4a8d24f9f0c3dccd18c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.632ex; height:2.676ex;" alt="{\displaystyle 7.35\times 10^{-3}}" /></span> with significand 7.35 and exponent −3.<sup id="cite_ref-122" class="reference"><a href="#cite_note-122"><span class="cite-bracket">[</span>117<span class="cite-bracket">]</span></a></sup> Unlike ordinary decimal notation, where trailing zeros of large numbers are implicitly considered to be non-significant, in scientific notation every digit in the significand is considered significant, and adding trailing zeros indicates higher precision. For example, while the number 1200 implicitly has only 2 significant digits, the number <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.20\times 10^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.20</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.20\times 10^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04a49e7be71902edd79865506cf4e5e2cd908eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.354ex; height:2.676ex;" alt="{\displaystyle 1.20\times 10^{3}}" /></span>⁠</span> explicitly has 3.<sup id="cite_ref-123" class="reference"><a href="#cite_note-123"><span class="cite-bracket">[</span>118<span class="cite-bracket">]</span></a></sup> </p><p>A common method employed by computers to approximate real number arithmetic is called <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>. It represents real numbers similar to the scientific notation through three numbers: a significand, a base, and an exponent.<sup id="cite_ref-124" class="reference"><a href="#cite_note-124"><span class="cite-bracket">[</span>119<span class="cite-bracket">]</span></a></sup> The precision of the significand is limited by the number of bits allocated to represent it. If an arithmetic operation results in a number that requires more bits than are available, the computer rounds the result to the closest representable number. This leads to <a href="/wiki/Rounding_error" class="mw-redirect" title="Rounding error">rounding errors</a>.<sup id="cite_ref-125" class="reference"><a href="#cite_note-125"><span class="cite-bracket">[</span>120<span class="cite-bracket">]</span></a></sup> A consequence of this behavior is that certain laws of arithmetic are violated by floating-point arithmetic. For example, floating-point addition is not associative since the rounding errors introduced can depend on the order of the additions. This means that the result of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7d8f6c490735eb0d2df7dd1d14fe0e2d9022ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle (a+b)+c}" /></span> is sometimes different from the result of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50291c75e2aebffcb9a27239b50d10f578ba8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle a+(b+c)}" /></span>.</span><sup id="cite_ref-126" class="reference"><a href="#cite_note-126"><span class="cite-bracket">[</span>121<span class="cite-bracket">]</span></a></sup> The most common technical standard used for floating-point arithmetic is called <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>. Among other things, it determines how numbers are represented, how arithmetic operations and rounding are performed, and how errors and exceptions are handled.<sup id="cite_ref-127" class="reference"><a href="#cite_note-127"><span class="cite-bracket">[</span>122<span class="cite-bracket">]</span></a></sup> In cases where computation speed is not a limiting factor, it is possible to use <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrary-precision arithmetic</a>, for which the precision of calculations is only restricted by the computer's memory.<sup id="cite_ref-128" class="reference"><a href="#cite_note-128"><span class="cite-bracket">[</span>123<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tool_use">Tool use</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=15" title="Edit section: Tool use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mental_calculation_at_primary_school.jpg" class="mw-file-description"><img alt="Painting of students engaged in mental arithmetic" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Mental_calculation_at_primary_school.jpg/220px-Mental_calculation_at_primary_school.jpg" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Mental_calculation_at_primary_school.jpg/330px-Mental_calculation_at_primary_school.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Mental_calculation_at_primary_school.jpg/440px-Mental_calculation_at_primary_school.jpg 2x" data-file-width="1283" data-file-height="958" /></a><figcaption>Calculations in <a href="/wiki/Mental_arithmetic" class="mw-redirect" title="Mental arithmetic">mental arithmetic</a> are done exclusively in the mind without relying on external aids.</figcaption></figure> <p>Forms of arithmetic can also be distinguished by the <a href="/wiki/Mathematical_instrument" title="Mathematical instrument">tools</a> employed to perform calculations and include many approaches besides the regular use of pen and paper. <a href="/wiki/Mental_arithmetic" class="mw-redirect" title="Mental arithmetic">Mental arithmetic</a> relies exclusively on the <a href="/wiki/Mind" title="Mind">mind</a> without external tools. Instead, it utilizes visualization, memorization, and certain calculation techniques to solve arithmetic problems.<sup id="cite_ref-129" class="reference"><a href="#cite_note-129"><span class="cite-bracket">[</span>124<span class="cite-bracket">]</span></a></sup> One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 85-47}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>85</mn> <mo>−</mo> <mn>47</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 85-47}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec30b07610091908d287f893e38106bb7c43f4f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.49ex; height:2.343ex;" alt="{\displaystyle 85-47}" /></span>, one calculates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 85-50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>85</mn> <mo>−</mo> <mn>50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 85-50}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f4d95ffe22c444668749bcf4950e8907aad42b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.49ex; height:2.343ex;" alt="{\displaystyle 85-50}" /></span> which is easier because it uses a round number. In the next step, one adds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" /></span> to the result to compensate for the earlier adjustment.<sup id="cite_ref-130" class="reference"><a href="#cite_note-130"><span class="cite-bracket">[</span>125<span class="cite-bracket">]</span></a></sup> Mental arithmetic is often taught in primary education to train the numerical abilities of the students.<sup id="cite_ref-131" class="reference"><a href="#cite_note-131"><span class="cite-bracket">[</span>126<span class="cite-bracket">]</span></a></sup> </p><p>The human body can also be employed as an arithmetic tool. The use of hands in <a href="/wiki/Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a> is often introduced to young children to teach them numbers and simple calculations. In its most basic form, the number of extended fingers corresponds to the represented quantity and arithmetic operations like addition and subtraction are performed by extending or retracting fingers. This system is limited to small numbers compared to more advanced systems which employ different approaches to represent larger quantities.<sup id="cite_ref-132" class="reference"><a href="#cite_note-132"><span class="cite-bracket">[</span>127<span class="cite-bracket">]</span></a></sup> The human voice is used as an arithmetic aid in verbal counting.<sup id="cite_ref-133" class="reference"><a href="#cite_note-133"><span class="cite-bracket">[</span>128<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Chinese-abacus.jpg" class="mw-file-description"><img alt="Photo of a Chinese abacus" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Chinese-abacus.jpg/220px-Chinese-abacus.jpg" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Chinese-abacus.jpg/330px-Chinese-abacus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Chinese-abacus.jpg/440px-Chinese-abacus.jpg 2x" data-file-width="2588" data-file-height="1700" /></a><figcaption>Abacuses are tools to perform arithmetic operations by moving beads.</figcaption></figure> <p><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a> are a simple system based on external tools other than the body. This system relies on mark making, such as strokes drawn on a surface or <a href="/wiki/Notch_(engineering)" title="Notch (engineering)">notches</a> carved into a wooden stick, to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read.<sup id="cite_ref-134" class="reference"><a href="#cite_note-134"><span class="cite-bracket">[</span>129<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Abacus" title="Abacus">abacus</a> is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several <a href="/wiki/Bead" title="Bead">beads</a>. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result.<sup id="cite_ref-135" class="reference"><a href="#cite_note-135"><span class="cite-bracket">[</span>130<span class="cite-bracket">]</span></a></sup> Related aids include <a href="/wiki/Counting_board" title="Counting board">counting boards</a>, which use tokens whose value depends on the area on the board in which they are placed,<sup id="cite_ref-136" class="reference"><a href="#cite_note-136"><span class="cite-bracket">[</span>131<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Counting_rods" title="Counting rods">counting rods</a>, which are arranged in horizontal and vertical patterns to represent different numbers.<sup id="cite_ref-137" class="reference"><a href="#cite_note-137"><span class="cite-bracket">[</span>132<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-139" class="reference"><a href="#cite_note-139"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Sector_(instrument)" title="Sector (instrument)">Sectors</a> and <a href="/wiki/Slide_rule" title="Slide rule">slide rules</a> are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations.<sup id="cite_ref-140" class="reference"><a href="#cite_note-140"><span class="cite-bracket">[</span>134<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-142" class="reference"><a href="#cite_note-142"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup> Printed tables were particularly relevant as an aid to look up the results of operations like logarithm and <a href="/wiki/Trigonometry" title="Trigonometry">trigonometric functions</a>.<sup id="cite_ref-143" class="reference"><a href="#cite_note-143"><span class="cite-bracket">[</span>136<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Mechanical_calculator" title="Mechanical calculator">Mechanical calculators</a> automate manual calculation processes. They present the user with some form of input device to enter numbers by turning dials or pressing keys. They include an internal mechanism usually consisting of <a href="/wiki/Gear" title="Gear">gears</a>, <a href="/wiki/Lever" title="Lever">levers</a>, and <a href="/wiki/Wheel" title="Wheel">wheels</a> to perform calculations and display the results.<sup id="cite_ref-144" class="reference"><a href="#cite_note-144"><span class="cite-bracket">[</span>137<span class="cite-bracket">]</span></a></sup> For <a href="/wiki/Electronic_calculator" class="mw-redirect" title="Electronic calculator">electronic calculators</a> and <a href="/wiki/Computer" title="Computer">computers</a>, this procedure is further refined by replacing the mechanical components with <a href="/wiki/Electronic_circuits" class="mw-redirect" title="Electronic circuits">electronic circuits</a> like <a href="/wiki/Microprocessor" title="Microprocessor">microprocessors</a> that combine and transform electric signals to perform calculations.<sup id="cite_ref-145" class="reference"><a href="#cite_note-145"><span class="cite-bracket">[</span>138<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Others">Others</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=16" title="Edit section: Others"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Clock_group.svg" class="mw-file-description"><img alt="Diagram of modular arithmetic using a clock" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/220px-Clock_group.svg.png" decoding="async" width="220" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/330px-Clock_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/440px-Clock_group.svg.png 2x" data-file-width="560" data-file-height="245" /></a><figcaption>Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock.</figcaption></figure> <p>There are many other types of arithmetic. <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a> operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division.<sup id="cite_ref-146" class="reference"><a href="#cite_note-146"><span class="cite-bracket">[</span>139<span class="cite-bracket">]</span></a></sup> </p><p>Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals.<sup id="cite_ref-147" class="reference"><a href="#cite_note-147"><span class="cite-bracket">[</span>140<span class="cite-bracket">]</span></a></sup> Vector arithmetic and matrix arithmetic describe arithmetic operations on <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, like <a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a> and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>.<sup id="cite_ref-148" class="reference"><a href="#cite_note-148"><span class="cite-bracket">[</span>141<span class="cite-bracket">]</span></a></sup> </p><p>Arithmetic systems can be classified based on the numeral system they rely on. For instance, <a href="/wiki/Decimal" title="Decimal">decimal</a> arithmetic describes arithmetic operations in the decimal system. Other examples are <a href="/wiki/Binary_number" title="Binary number">binary</a> arithmetic, <a href="/wiki/Octal" title="Octal">octal</a> arithmetic, and <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> arithmetic.<sup id="cite_ref-149" class="reference"><a href="#cite_note-149"><span class="cite-bracket">[</span>142<span class="cite-bracket">]</span></a></sup> </p><p>Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min.<sup id="cite_ref-150" class="reference"><a href="#cite_note-150"><span class="cite-bracket">[</span>143<span class="cite-bracket">]</span></a></sup> </p><p>Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5120dbef0aa4b1760be59a33516661780dfc7a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=1}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+2=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4520e2da0b35c4bf907b9d6b933fb015ea0e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2+2=5}" /></span>.<sup id="cite_ref-151" class="reference"><a href="#cite_note-151"><span class="cite-bracket">[</span>144<span class="cite-bracket">]</span></a></sup> They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5120dbef0aa4b1760be59a33516661780dfc7a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=1}" /></span> can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one.<sup id="cite_ref-152" class="reference"><a href="#cite_note-152"><span class="cite-bracket">[</span>145<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Axiomatic_foundations">Axiomatic foundations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=17" title="Edit section: Axiomatic foundations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic foundations</a> of arithmetic try to provide a small set of laws, called <a href="/wiki/Axiom" title="Axiom">axioms</a>, from which all fundamental properties of and operations on numbers can be derived. They constitute logically consistent and systematic frameworks that can be used to formulate <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proofs</a> in a rigorous manner. Two well-known approaches are the <a href="/wiki/Dedekind%E2%80%93Peano_axioms" class="mw-redirect" title="Dedekind–Peano axioms">Dedekind–Peano axioms</a> and <a href="/wiki/Set-theoretic" class="mw-redirect" title="Set-theoretic">set-theoretic</a> constructions.<sup id="cite_ref-153" class="reference"><a href="#cite_note-153"><span class="cite-bracket">[</span>146<span class="cite-bracket">]</span></a></sup> </p><p>The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> and later refined by <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and <a href="/wiki/Successor_function" title="Successor function">successor</a>.<sup id="cite_ref-154" class="reference"><a href="#cite_note-154"><span class="cite-bracket">[</span>h<span class="cite-bracket">]</span></a></sup> The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts.<sup id="cite_ref-155" class="reference"><a href="#cite_note-155"><span class="cite-bracket">[</span>147<span class="cite-bracket">]</span></a></sup> </p> <ul><li>0 is a natural number.</li> <li>For every natural number, there is a successor, which is also a natural number.</li> <li>The successors of two different natural numbers are never identical.</li> <li>0 is not the successor of a natural number.</li> <li>If a set contains 0 and every successor then it contains every natural number.<sup id="cite_ref-156" class="reference"><a href="#cite_note-156"><span class="cite-bracket">[</span>148<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-158" class="reference"><a href="#cite_note-158"><span class="cite-bracket">[</span>i<span class="cite-bracket">]</span></a></sup></li></ul> <p>Numbers greater than 0 are expressed by repeated application of the successor function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de385dbef3a40036f2b9869ad29c9a77cb4e663e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.062ex; height:2.843ex;" alt="{\displaystyle s(0)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(s(s(0)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(s(s(0)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f627269803b2cd02eeda2623b08917a4b00bd12c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.862ex; height:2.843ex;" alt="{\displaystyle s(s(s(0)))}" /></span>. Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" /></span> to any number is the same as applying the successor function two times to this number.<sup id="cite_ref-159" class="reference"><a href="#cite_note-159"><span class="cite-bracket">[</span>150<span class="cite-bracket">]</span></a></sup> </p><p>Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }" /></span>. Each subsequent number can be defined as the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of the previous number with the set containing the previous number. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=0\cup \{0\}=\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=0\cup \{0\}=\{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39287abe1ae51131b05c87afadd39cf858a07d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.079ex; height:2.843ex;" alt="{\displaystyle 1=0\cup \{0\}=\{0\}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2=1\cup \{1\}=\{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mn>1</mn> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2=1\cup \{1\}=\{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea5731586a681ff59f141589a589244a4492d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.276ex; height:2.843ex;" alt="{\displaystyle 2=1\cup \{1\}=\{0,1\}}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3=2\cup \{2\}=\{0,1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>=</mo> <mn>2</mn> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3=2\cup \{2\}=\{0,1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd26842d1630f8758f36cdfa92536fcef9776f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.472ex; height:2.843ex;" alt="{\displaystyle 3=2\cup \{2\}=\{0,1,2\}}" /></span>.<sup id="cite_ref-160" class="reference"><a href="#cite_note-160"><span class="cite-bracket">[</span>151<span class="cite-bracket">]</span></a></sup> Integers can be defined as <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number -9.<sup id="cite_ref-161" class="reference"><a href="#cite_note-161"><span class="cite-bracket">[</span>152<span class="cite-bracket">]</span></a></sup> Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f2b824decf224d9a3143d4271666c7fba7ac83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{7}}}" /></span>.<sup id="cite_ref-162" class="reference"><a href="#cite_note-162"><span class="cite-bracket">[</span>153<span class="cite-bracket">]</span></a></sup> One way to construct the real numbers relies on the concept of <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cuts</a>. According to this approach, each real number is represented by a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest.<sup id="cite_ref-163" class="reference"><a href="#cite_note-163"><span class="cite-bracket">[</span>154<span class="cite-bracket">]</span></a></sup> Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result.<sup id="cite_ref-164" class="reference"><a href="#cite_note-164"><span class="cite-bracket">[</span>155<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=18" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ishango_bone_(cropped).jpg" class="mw-file-description"><img alt="Photo of the Ishango bone" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/130px-Ishango_bone_%28cropped%29.jpg" decoding="async" width="130" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/195px-Ishango_bone_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/260px-Ishango_bone_%28cropped%29.jpg 2x" data-file-width="891" data-file-height="1230" /></a><figcaption>Some historians interpret the <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a> as one of the earliest arithmetic artifacts.</figcaption></figure> <p>The earliest forms of arithmetic are sometimes traced back to <a href="/wiki/Counting" title="Counting">counting</a> and <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a> used to keep track of quantities. Some historians suggest that the <a href="/wiki/Lebombo_bone" title="Lebombo bone">Lebombo bone</a> (dated about 43,000 years ago) and the <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a> (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.<sup id="cite_ref-165" class="reference"><a href="#cite_note-165"><span class="cite-bracket">[</span>156<span class="cite-bracket">]</span></a></sup> However, a basic <a href="/wiki/Number_sense" title="Number sense">sense of numbers</a> may predate these findings and might even have existed before the development of language.<sup id="cite_ref-166" class="reference"><a href="#cite_note-166"><span class="cite-bracket">[</span>157<span class="cite-bracket">]</span></a></sup> </p><p>It was not until the emergence of <a href="/wiki/Ancient_civilizations" class="mw-redirect" title="Ancient civilizations">ancient civilizations</a> that a more complex and structured approach to arithmetic began to evolve, starting around 3000 BCE. This became necessary because of the increased need to keep track of stored items, manage land ownership, and arrange exchanges.<sup id="cite_ref-167" class="reference"><a href="#cite_note-167"><span class="cite-bracket">[</span>158<span class="cite-bracket">]</span></a></sup> All the major ancient civilizations developed non-positional numeral systems to facilitate the representation of numbers. They also had symbols for operations like addition and subtraction and were aware of fractions. Examples are <a href="/wiki/Egyptian_hieroglyphics" class="mw-redirect" title="Egyptian hieroglyphics">Egyptian hieroglyphics</a> as well as the numeral systems invented in <a href="/wiki/Sumeria" class="mw-redirect" title="Sumeria">Sumeria</a>, <a href="/wiki/Ancient_China" class="mw-redirect" title="Ancient China">China</a>, and <a href="/wiki/Ancient_India" class="mw-redirect" title="Ancient India">India</a>.<sup id="cite_ref-168" class="reference"><a href="#cite_note-168"><span class="cite-bracket">[</span>159<span class="cite-bracket">]</span></a></sup> The first <a href="/wiki/Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral system</a> was developed by the <a href="/wiki/Babylonians" class="mw-redirect" title="Babylonians">Babylonians</a> starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient.<sup id="cite_ref-169" class="reference"><a href="#cite_note-169"><span class="cite-bracket">[</span>160<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Abacus" title="Abacus">Abacuses</a> have been utilized as hand-operated calculating tools since ancient times as efficient means for performing complex calculations.<sup id="cite_ref-170" class="reference"><a href="#cite_note-170"><span class="cite-bracket">[</span>161<span class="cite-bracket">]</span></a></sup> </p><p>Early civilizations primarily used numbers for concrete practical purposes, like commercial activities and tax records, but lacked an abstract concept of number itself.<sup id="cite_ref-171" class="reference"><a href="#cite_note-171"><span class="cite-bracket">[</span>162<span class="cite-bracket">]</span></a></sup> This changed with the <a href="/wiki/Ancient_Greek_mathematicians" class="mw-redirect" title="Ancient Greek mathematicians">ancient Greek mathematicians</a>, who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems.<sup id="cite_ref-172" class="reference"><a href="#cite_note-172"><span class="cite-bracket">[</span>163<span class="cite-bracket">]</span></a></sup> Another novel feature was their use of <a href="/wiki/Mathematical_proof" title="Mathematical proof">proofs</a> to establish mathematical truths and validate theories.<sup id="cite_ref-173" class="reference"><a href="#cite_note-173"><span class="cite-bracket">[</span>164<span class="cite-bracket">]</span></a></sup> A further contribution was their distinction of various classes of numbers, such as <a href="/wiki/Even_numbers" class="mw-redirect" title="Even numbers">even numbers</a>, odd numbers, and <a href="/wiki/Prime_numbers" class="mw-redirect" title="Prime numbers">prime numbers</a>.<sup id="cite_ref-174" class="reference"><a href="#cite_note-174"><span class="cite-bracket">[</span>165<span class="cite-bracket">]</span></a></sup> This included the discovery that numbers for certain geometrical lengths are <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> and therefore cannot be expressed as a fraction.<sup id="cite_ref-175" class="reference"><a href="#cite_note-175"><span class="cite-bracket">[</span>166<span class="cite-bracket">]</span></a></sup> The works of <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> and <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics.<sup id="cite_ref-176" class="reference"><a href="#cite_note-176"><span class="cite-bracket">[</span>167<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> was an influential figure in Greek arithmetic in the 3rd century BCE because of his numerous contributions to <a href="/wiki/Number_theory" title="Number theory">number theory</a> and his exploration of the application of arithmetic operations to <a href="/wiki/Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a>.<sup id="cite_ref-177" class="reference"><a href="#cite_note-177"><span class="cite-bracket">[</span>168<span class="cite-bracket">]</span></a></sup> </p><p>The ancient Indians were the first to develop the concept of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> as a number to be used in calculations. The exact rules of its operation were written down by <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> in around 628 CE.<sup id="cite_ref-178" class="reference"><a href="#cite_note-178"><span class="cite-bracket">[</span>169<span class="cite-bracket">]</span></a></sup> The concept of zero or none existed long before, but it was not considered an object of arithmetic operations.<sup id="cite_ref-179" class="reference"><a href="#cite_note-179"><span class="cite-bracket">[</span>170<span class="cite-bracket">]</span></a></sup> Brahmagupta further provided a detailed discussion of calculations with <a href="/wiki/Negative_numbers" class="mw-redirect" title="Negative numbers">negative numbers</a> and their application to problems like credit and debt.<sup id="cite_ref-180" class="reference"><a href="#cite_note-180"><span class="cite-bracket">[</span>171<span class="cite-bracket">]</span></a></sup> The concept of negative numbers itself is significantly older and was <a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">first explored</a> in <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematics</a> in the first millennium BCE.<sup id="cite_ref-181" class="reference"><a href="#cite_note-181"><span class="cite-bracket">[</span>172<span class="cite-bracket">]</span></a></sup> </p><p>Indian mathematicians also developed the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">positional decimal</a> system used today, in particular the concept of a zero digit instead of empty or missing positions.<sup id="cite_ref-182" class="reference"><a href="#cite_note-182"><span class="cite-bracket">[</span>173<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Aryabhatiya" title="Aryabhatiya">a detailed treatment</a> of its operations was provided by <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> around the turn of the 6th century CE.<sup id="cite_ref-183" class="reference"><a href="#cite_note-183"><span class="cite-bracket">[</span>174<span class="cite-bracket">]</span></a></sup> The Indian decimal system was further refined and expanded to non-integers during the <a href="/wiki/Islamic_Golden_Age" title="Islamic Golden Age">Islamic Golden Age</a> by Middle Eastern mathematicians such as <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a>. His work was influential in introducing the decimal numeral system to the Western world, which at that time relied on the <a href="/wiki/Roman_numeral_system" class="mw-redirect" title="Roman numeral system">Roman numeral system</a>.<sup id="cite_ref-184" class="reference"><a href="#cite_note-184"><span class="cite-bracket">[</span>175<span class="cite-bracket">]</span></a></sup> There, it was popularized by mathematicians like <a href="/wiki/Leonardo_Fibonacci" class="mw-redirect" title="Leonardo Fibonacci">Leonardo Fibonacci</a>, who lived in the 12th and 13th centuries and also developed the <a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a>.<sup id="cite_ref-185" class="reference"><a href="#cite_note-185"><span class="cite-bracket">[</span>176<span class="cite-bracket">]</span></a></sup> During the <a href="/wiki/Middle_Ages" title="Middle Ages">Middle Ages</a> and <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, many popular textbooks were published to cover the practical calculations for commerce. The use of abacuses also became widespread in this period.<sup id="cite_ref-186" class="reference"><a href="#cite_note-186"><span class="cite-bracket">[</span>177<span class="cite-bracket">]</span></a></sup> In the 16th century, the mathematician <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> conceived the concept of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> as a way to solve <a href="/wiki/Cubic_equations" class="mw-redirect" title="Cubic equations">cubic equations</a>.<sup id="cite_ref-187" class="reference"><a href="#cite_note-187"><span class="cite-bracket">[</span>178<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Leibniz_Stepped_Reckoner.png" class="mw-file-description"><img alt="Photo of Leibniz's stepped reckoner" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Leibniz_Stepped_Reckoner.png/220px-Leibniz_Stepped_Reckoner.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Leibniz_Stepped_Reckoner.png/330px-Leibniz_Stepped_Reckoner.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Leibniz_Stepped_Reckoner.png/440px-Leibniz_Stepped_Reckoner.png 2x" data-file-width="1235" data-file-height="803" /></a><figcaption>Leibniz's <a href="/wiki/Stepped_reckoner" title="Stepped reckoner">stepped reckoner</a> was the first calculator that could perform all four arithmetic operations.<sup id="cite_ref-188" class="reference"><a href="#cite_note-188"><span class="cite-bracket">[</span>179<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The first <a href="/wiki/Mechanical_calculator" title="Mechanical calculator">mechanical calculators</a> were developed in the 17th century and greatly facilitated complex mathematical calculations, such as <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>'s <a href="/wiki/Pascal%27s_calculator" class="mw-redirect" title="Pascal's calculator">calculator</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>'s <a href="/wiki/Stepped_reckoner" title="Stepped reckoner">stepped reckoner</a>.<sup id="cite_ref-189" class="reference"><a href="#cite_note-189"><span class="cite-bracket">[</span>180<span class="cite-bracket">]</span></a></sup> The 17th century also saw the discovery of the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> by <a href="/wiki/John_Napier" title="John Napier">John Napier</a>.<sup id="cite_ref-190" class="reference"><a href="#cite_note-190"><span class="cite-bracket">[</span>181<span class="cite-bracket">]</span></a></sup> </p><p>In the 18th and 19th centuries, mathematicians such as <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> and <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> laid the foundations of modern number theory.<sup id="cite_ref-191" class="reference"><a href="#cite_note-191"><span class="cite-bracket">[</span>182<span class="cite-bracket">]</span></a></sup> Another development in this period concerned work on the formalization and foundations of arithmetic, such as <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s <a href="/wiki/Set_theory" title="Set theory">set theory</a> and the <a href="/wiki/Dedekind%E2%80%93Peano_axioms" class="mw-redirect" title="Dedekind–Peano axioms">Dedekind–Peano axioms</a> used as an axiomatization of natural-number arithmetic.<sup id="cite_ref-192" class="reference"><a href="#cite_note-192"><span class="cite-bracket">[</span>183<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Computers" class="mw-redirect" title="Computers">Computers</a> and <a href="/wiki/Electronic_calculators" class="mw-redirect" title="Electronic calculators">electronic calculators</a> were first developed in the 20th century. Their widespread use revolutionized both the accuracy and speed with which even complex arithmetic computations can be calculated.<sup id="cite_ref-193" class="reference"><a href="#cite_note-193"><span class="cite-bracket">[</span>184<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_various_fields">In various fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=19" title="Edit section: In various fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Education">Education</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=20" title="Edit section: Education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></div> <p>Arithmetic education forms part of <a href="/wiki/Primary_education" title="Primary education">primary education</a>. It is one of the first forms of <a href="/wiki/Mathematics_education" title="Mathematics education">mathematics education</a> that children encounter. <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">Elementary arithmetic</a> aims to give students a basic <a href="/wiki/Number_sense" title="Number sense">sense of numbers</a> and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division.<sup id="cite_ref-194" class="reference"><a href="#cite_note-194"><span class="cite-bracket">[</span>185<span class="cite-bracket">]</span></a></sup> It is usually introduced in relation to concrete scenarios, like counting <a href="/wiki/Bead" title="Bead">beads</a>, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in early arithmetic education are <a href="/wiki/Number_lines" class="mw-redirect" title="Number lines">number lines</a>, addition and multiplication tables, <a href="/wiki/Counting_blocks" class="mw-redirect" title="Counting blocks">counting blocks</a>, and abacuses.<sup id="cite_ref-195" class="reference"><a href="#cite_note-195"><span class="cite-bracket">[</span>186<span class="cite-bracket">]</span></a></sup> </p><p>Later stages focus on a more abstract understanding and introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers. They further cover more advanced numerical operations, like exponentiation, extraction of roots, and logarithm.<sup id="cite_ref-196" class="reference"><a href="#cite_note-196"><span class="cite-bracket">[</span>187<span class="cite-bracket">]</span></a></sup> They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> and <a href="/wiki/Calculator" title="Calculator">calculators</a> to solve complex arithmetic problems.<sup id="cite_ref-197" class="reference"><a href="#cite_note-197"><span class="cite-bracket">[</span>188<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Psychology">Psychology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=21" title="Edit section: Psychology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Psychology" title="Psychology">psychology</a> of arithmetic is interested in how humans and animals learn about numbers, represent them, and use them for calculations. It examines how mathematical problems are understood and solved and how arithmetic abilities are related to <a href="/wiki/Perception" title="Perception">perception</a>, <a href="/wiki/Memory" title="Memory">memory</a>, <a href="/wiki/Judgment" class="mw-redirect" title="Judgment">judgment</a>, and <a href="/wiki/Decision_making" class="mw-redirect" title="Decision making">decision making</a>.<sup id="cite_ref-198" class="reference"><a href="#cite_note-198"><span class="cite-bracket">[</span>189<span class="cite-bracket">]</span></a></sup> For example, it investigates how collections of concrete items are first encountered in perception and subsequently associated with numbers.<sup id="cite_ref-199" class="reference"><a href="#cite_note-199"><span class="cite-bracket">[</span>190<span class="cite-bracket">]</span></a></sup> A further field of inquiry concerns the relation between numerical calculations and the use of language to form representations.<sup id="cite_ref-200" class="reference"><a href="#cite_note-200"><span class="cite-bracket">[</span>191<span class="cite-bracket">]</span></a></sup> Psychology also explores the biological origin of arithmetic as an inborn ability. This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation.<sup id="cite_ref-201" class="reference"><a href="#cite_note-201"><span class="cite-bracket">[</span>192<span class="cite-bracket">]</span></a></sup> </p><p>One of the concepts studied by psychology is <a href="/wiki/Numeracy" title="Numeracy">numeracy</a>, which is the capability to comprehend numerical concepts, apply them to concrete situations, and <a href="/wiki/Logical_reasoning" title="Logical reasoning">reason</a> with them. It includes a fundamental number sense as well as being able to estimate and compare quantities. It further encompasses the abilities to symbolically represent numbers in numbering systems, interpret <a href="/wiki/Numerical_data" class="mw-redirect" title="Numerical data">numerical data</a>, and evaluate arithmetic calculations.<sup id="cite_ref-202" class="reference"><a href="#cite_note-202"><span class="cite-bracket">[</span>193<span class="cite-bracket">]</span></a></sup> Numeracy is a key skill in many academic fields. A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, by misunderstanding <a href="/wiki/Mortgage" title="Mortgage">mortgage</a> plans and <a href="/wiki/Insurance_policies" class="mw-redirect" title="Insurance policies">insurance policies</a>.<sup id="cite_ref-203" class="reference"><a href="#cite_note-203"><span class="cite-bracket">[</span>194<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Philosophy">Philosophy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=22" title="Edit section: Philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></div> <p>The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and <a href="/wiki/Ontology" title="Ontology">ontological status</a> of numbers, the relation of arithmetic to language and <a href="/wiki/Logic" title="Logic">logic</a>, and how it is possible to acquire arithmetic <a href="/wiki/Knowledge" title="Knowledge">knowledge</a>.<sup id="cite_ref-204" class="reference"><a href="#cite_note-204"><span class="cite-bracket">[</span>195<span class="cite-bracket">]</span></a></sup> </p><p>According to <a href="/wiki/Platonism" title="Platonism">Platonism</a>, numbers have mind-independent existence: they exist as <a href="/wiki/Abstract_objects" class="mw-redirect" title="Abstract objects">abstract objects</a> outside spacetime and without causal powers.<sup id="cite_ref-205" class="reference"><a href="#cite_note-205"><span class="cite-bracket">[</span>196<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-207" class="reference"><a href="#cite_note-207"><span class="cite-bracket">[</span>j<span class="cite-bracket">]</span></a></sup> This view is rejected by <a href="/wiki/Intuitionism" title="Intuitionism">intuitionists</a>, who claim that mathematical objects are mental constructions.<sup id="cite_ref-208" class="reference"><a href="#cite_note-208"><span class="cite-bracket">[</span>198<span class="cite-bracket">]</span></a></sup> Further theories are <a href="/wiki/Logicism" title="Logicism">logicism</a>, which holds that mathematical truths are reducible to <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a>,<sup id="cite_ref-209" class="reference"><a href="#cite_note-209"><span class="cite-bracket">[</span>199<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalism</a>, which states that mathematical principles are rules of how symbols are manipulated without claiming that they correspond to entities outside the rule-governed activity.<sup id="cite_ref-210" class="reference"><a href="#cite_note-210"><span class="cite-bracket">[</span>200<span class="cite-bracket">]</span></a></sup> </p><p>The traditionally dominant view in the <a href="/wiki/Epistemology" title="Epistemology">epistemology</a> of arithmetic is that arithmetic truths are knowable <a href="/wiki/A_priori" class="mw-redirect" title="A priori">a priori</a>. This means that they can be known by thinking alone without the need to rely on <a href="/wiki/Sensory_experience" class="mw-redirect" title="Sensory experience">sensory experience</a>.<sup id="cite_ref-211" class="reference"><a href="#cite_note-211"><span class="cite-bracket">[</span>201<span class="cite-bracket">]</span></a></sup> Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of <a href="/wiki/Rational_intuition" class="mw-redirect" title="Rational intuition">rational intuition</a> through which mathematical truths can be apprehended.<sup id="cite_ref-212" class="reference"><a href="#cite_note-212"><span class="cite-bracket">[</span>202<span class="cite-bracket">]</span></a></sup> A more recent alternative view was suggested by <a href="/wiki/Naturalism_(philosophy)" title="Naturalism (philosophy)">naturalist</a> philosophers like <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a>, who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences.<sup id="cite_ref-213" class="reference"><a href="#cite_note-213"><span class="cite-bracket">[</span>203<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Others_2">Others</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=23" title="Edit section: Others"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Arithmetic is relevant to many fields. In <a href="/wiki/Daily_life" class="mw-redirect" title="Daily life">daily life</a>, it is required to calculate change when shopping, manage <a href="/wiki/Personal_finances" class="mw-redirect" title="Personal finances">personal finances</a>, and adjust a cooking recipe for a different number of servings. Businesses use arithmetic to calculate profits and losses and <a href="/wiki/Marketing_research" title="Marketing research">analyze market trends</a>. In the field of <a href="/wiki/Engineering" title="Engineering">engineering</a>, it is used to measure quantities, calculate loads and forces, and design structures.<sup id="cite_ref-214" class="reference"><a href="#cite_note-214"><span class="cite-bracket">[</span>204<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> relies on arithmetic operations to protect sensitive information by <a href="/wiki/Encrypt" class="mw-redirect" title="Encrypt">encrypting</a> data and messages.<sup id="cite_ref-215" class="reference"><a href="#cite_note-215"><span class="cite-bracket">[</span>205<span class="cite-bracket">]</span></a></sup> </p><p>Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations. <a href="/wiki/Algebra" title="Algebra">Algebra</a> relies on arithmetic principles to solve <a href="/wiki/Equation" title="Equation">equations</a> using variables. These principles also play a key role in <a href="/wiki/Calculus" title="Calculus">calculus</a> in its attempt to determine <a href="/wiki/Rates_of_change" class="mw-redirect" title="Rates of change">rates of change</a> and areas under <a href="/wiki/Curve" title="Curve">curves</a>. <a href="/wiki/Geometry" title="Geometry">Geometry</a> uses arithmetic operations to measure the properties of shapes while <a href="/wiki/Statistics" title="Statistics">statistics</a> utilizes them to analyze numerical data.<sup id="cite_ref-216" class="reference"><a href="#cite_note-216"><span class="cite-bracket">[</span>206<span class="cite-bracket">]</span></a></sup> Due to the relevance of arithmetic operations throughout mathematics, the influence of arithmetic extends to most sciences such as <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, and <a href="/wiki/Economics" title="Economics">economics</a>. These operations are used in calculations, <a href="/wiki/Problem-solving" class="mw-redirect" title="Problem-solving">problem-solving</a>, <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>, and algorithms, making them integral to scientific research, technological development, and economic modeling.<sup id="cite_ref-217" class="reference"><a href="#cite_note-217"><span class="cite-bracket">[</span>207<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/28px-Arithmetic_symbols.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/42px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/56px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Arithmetic" title="Portal:Arithmetic">Arithmetic portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Algorism" title="Algorism">Algorism</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression (mathematics)</a></li> <li><a href="/wiki/Finite_field_arithmetic" title="Finite field arithmetic">Finite field arithmetic</a></li> <li><a href="/wiki/Outline_of_arithmetic" title="Outline of arithmetic">Outline of arithmetic</a></li> <li><a href="/wiki/Plant_arithmetic" title="Plant arithmetic">Plant arithmetic</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=26" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Other symbols for the natural numbers include <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aad06b66db15cb0bccc195143bafdb86c9afeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {N} ^{*}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a339dbd4a64016f4d222b5cd5d840d77041924a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ^{+}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cca915d54ae835781191ae19599e11c7ff3d066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{1}}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }" /></span>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Other symbols for the whole numbers include <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c266661ca769727f799db89ef85b1353f2b1d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{0}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \cup \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc48c7f4b68077eefe8ed39fb17e06ae15e3eccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.748ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \{0\}}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" /></span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">A repeating decimal is a decimal with an infinite number of repeating digits, like 0.111..., which expresses the rational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7fbb8c7af3dce2f4bb214f14a76358a32a49d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{9}}}" /></span>.</span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text">Some authors use a different terminology and refer to the first number as multiplicand and the second number as the multiplier.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text">If the exponent is 0 then the result is 1, as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d03eef927804acedd3fc1fc38ba64342017f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 7^{0}=1}" /></span>. The only exception is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106f0c4e1cbccbfcbb61001a8c17b8427c65366d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 0^{0}}" /></span>, which is not defined.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-139"><span class="mw-cite-backlink"><b><a href="#cite_ref-139">^</a></b></span> <span class="reference-text">Some systems of counting rods include different colors to represent both positive and negative numbers.<sup id="cite_ref-138" class="reference"><a href="#cite_note-138"><span class="cite-bracket">[</span>133<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-142"><span class="mw-cite-backlink"><b><a href="#cite_ref-142">^</a></b></span> <span class="reference-text">Some computer scientists see slide rules as the first type of <a href="/wiki/Analog_computer" title="Analog computer">analog computer</a>.<sup id="cite_ref-141" class="reference"><a href="#cite_note-141"><span class="cite-bracket">[</span>135<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-154"><span class="mw-cite-backlink"><b><a href="#cite_ref-154">^</a></b></span> <span class="reference-text">The successor of a natural number is the number that comes after it. For instance, 4 is the successor of 3.</span> </li> <li id="cite_note-158"><span class="mw-cite-backlink"><b><a href="#cite_ref-158">^</a></b></span> <span class="reference-text">There are different versions of the exact formulation and number of axioms. For example, some formulations start with 1 instead of 0 in the first axiom.<sup id="cite_ref-157" class="reference"><a href="#cite_note-157"><span class="cite-bracket">[</span>149<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-207"><span class="mw-cite-backlink"><b><a href="#cite_ref-207">^</a></b></span> <span class="reference-text">An <a href="/wiki/Quine%E2%80%93Putnam_indispensability_argument" title="Quine–Putnam indispensability argument">influential argument</a> for Platonism, first formulated by <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a> and <a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a>, states that numbers exist because they are indispensable to the best scientific theories.<sup id="cite_ref-FOOTNOTEColyvan2023Lead_Section_206-0" class="reference"><a href="#cite_note-FOOTNOTEColyvan2023Lead_Section-206"><span class="cite-bracket">[</span>197<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=27" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 21em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, pp. 302–303</li><li><a href="#CITEREFHC_staff2022b">HC staff 2022b</a></li><li><a href="#CITEREFMW_staff2023">MW staff 2023</a></li><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li></ul></div></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. 57, 77</li><li><a href="#CITEREFAdamowicz1994">Adamowicz 1994</a>, p. 299</li></ul></div></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPeirce2015">Peirce 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oI_yCQAAQBAJ&pg=PA109">109</a></li><li><a href="#CITEREFWaite2013">Waite 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xqKcAQAAQBAJ&pg=PA42">42</a></li><li><a href="#CITEREFSmith1958">Smith 1958</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC&pg=PA7">7</a></li></ul></div></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFHofweber2016">Hofweber 2016</a>, p. 153</li></ul></div></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, pp. 302–303</li><li><a href="#CITEREFHC_staff2022b">HC staff 2022b</a></li><li><a href="#CITEREFMW_staff2023">MW staff 2023</a></li><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li></ul></div></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFSophian2017">Sophian 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8FYPEAAAQBAJ&pg=PA84">84</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li><li><a href="#CITEREFStevensonWaite2011">Stevenson & Waite 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sYScAQAAQBAJ&pg=PA70">70</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, pp. 303–304</li></ul></div></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLozano-Robledo2019">Lozano-Robledo 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13">xiii</a></li><li><a href="#CITEREFNagelNewman2008">Nagel & Newman 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4">4</a></li></ul></div></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFWilson2020">Wilson 2020</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1">1–2</a></li><li><a href="#CITEREFKaratsuba2020">Karatsuba 2020</a></li><li><a href="#CITEREFCampbell2012">Campbell 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yoEFp-Q2OXIC&pg=PT33">33</a></li><li><a href="#CITEREFRobbins2006">Robbins 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1">1</a></li></ul></div></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFDuverney2010">Duverney 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sr5S9oN1xPAC&pg=PR5">v</a></li><li><a href="#CITEREFRobbins2006">Robbins 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1">1</a></li></ul></div></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, pp. 302–304</li><li><a href="#CITEREFKhattar2010">Khattar 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I3rCgXwvffsC&pg=PA1">1–2</a></li><li><a href="#CITEREFNakovKolev2013">Nakov & Kolev 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA270">270–271</a></li></ul></div></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFLudererNollauVetters2013">Luderer, Nollau & Vetters 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rSf0CAAAQBAJ&pg=PA9">9</a></li><li><a href="#CITEREFKhattar2010">Khattar 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I3rCgXwvffsC&pg=PA1">1–2</a></li></ul></div></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li><li><a href="#CITEREFZhang2012">Zhang 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GRTSBwAAQBAJ&pg=PA130">130</a></li><li><a href="#CITEREFKörner2019">Körner 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z-y2DwAAQBAJ&pg=PA109">109</a></li><li><a href="#CITEREFInternational_Organization_for_Standardization2019">International Organization for Standardization 2019</a>, p. <a rel="nofollow" class="external text" href="https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf">4</a></li></ul></div></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li></ul></div></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFSwanson2021">Swanson 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cHshEAAAQBAJ&pg=PA107">107</a></li><li><a href="#CITEREFRossi2011">Rossi 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kSwVGbBtel8C&pg=PA111">111</a></li></ul></div></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRajan2022">Rajan 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OCE6EAAAQBAJ&pg=PA17">17</a></li><li><a href="#CITEREFHafstrom2013">Hafstrom 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA6">6</a></li></ul></div></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFHafstrom2013">Hafstrom 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA95">95</a></li></ul></div></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOrr1995">Orr 1995</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DMTqRnoE8iMC&pg=PA49">49</a></li><li><a href="#CITEREFNelson2019">Nelson 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xTiDDwAAQBAJ&pg=PR31">xxxi</a></li></ul></div></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFHafstrom2013">Hafstrom 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA123">123</a></li></ul></div></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA33">33</a></li></ul></div></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358">358</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li></ul></div></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358">358–359</a></li><li><a href="#CITEREFRooney2021">Rooney 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34">34</a></li></ul></div></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li></ul></div></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFWard2012">Ward 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LVDvCAAAQBAJ&pg=PA55">55</a></li></ul></div></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 1–2</li><li><a href="#CITEREFHC_staff2022">HC staff 2022</a></li><li><a href="#CITEREFHC_staff2022a">HC staff 2022a</a></li></ul></div></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 8–10</li><li><a href="#CITEREFNakovKolev2013">Nakov & Kolev 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA270">270–272</a></li></ul></div></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFStakhov2020">Stakhov 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA73">73</a></li><li><a href="#CITEREFNakovKolev2013">Nakov & Kolev 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA271">271–272</a></li><li><a href="#CITEREFJena2021">Jena 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17">17–18</a></li></ul></div></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNakovKolev2013">Nakov & Kolev 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA271">271–272</a></li><li><a href="#CITEREFJena2021">Jena 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17">17–18</a></li></ul></div></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 8–10</li><li><a href="#CITEREFMazumderEbong2023">Mazumder & Ebong 2023</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18">18–19</a></li><li><a href="#CITEREFMoncayo2018">Moncayo 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J-pTDwAAQBAJ&pg=PT25">25</a></li></ul></div></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, p. 8</li><li><a href="#CITEREFMazumderEbong2023">Mazumder & Ebong 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18">18</a></li></ul></div></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a href="#CITEREFOre1948">Ore 1948</a>, p. 10</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 8–10</li><li><a href="#CITEREFMazumderEbong2023">Mazumder & Ebong 2023</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18">18–19</a></li><li><a href="#CITEREFStakhov2020">Stakhov 2020</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA77">77–78</a></li></ul></div></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFYan2002">Yan 2002</a>, pp. 305–306</li><li><a href="#CITEREFITL_Education_Solutions_Limited2011">ITL Education Solutions Limited 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28">28</a></li><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 2–3</li><li><a href="#CITEREFJena2021">Jena 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17">17–18</a></li></ul></div></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNagel2002">Nagel 2002</a>, p. 178</li><li><a href="#CITEREFJena2021">Jena 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA20">20–21</a></li><li><a href="#CITEREFNullLobur2006">Null & Lobur 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QGPHAl9GE-IC&pg=PA40">40</a></li></ul></div></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="#CITEREFStakhov2020">Stakhov 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA74">74</a></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNagel2002">Nagel 2002</a>, p. 179</li><li><a href="#CITEREFHusserlWillard2012">Husserl & Willard 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lxftCAAAQBAJ&pg=PR44">XLIV–XLV</a></li><li><a href="#CITEREFO'Leary2015">O'Leary 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ci6kBgAAQBAJ&pg=PA190">190</a></li></ul></div></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRisingMatthewsSchoaffMatthew2021">Rising et al. 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hjVZEAAAQBAJ&pg=PA110">110</a></li><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 177, 179–180</li></ul></div></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. 57, 77</li><li><a href="#CITEREFAdamowicz1994">Adamowicz 1994</a>, p. 299</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 177, 179–180</li></ul></div></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKhanGraham2018">Khan & Graham 2018</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vy73DwAAQBAJ&pg=PA9">9–10</a></li><li><a href="#CITEREFSmyth1864">Smyth 1864</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BqQZAAAAYAAJ&pg=PA55">55</a></li></ul></div></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFTarasov2008">Tarasov 2008</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pHK11tfdE3QC&pg=PA57">57–58</a></li><li><a href="#CITEREFMazzolaMilmeisterWeissmann2004">Mazzola, Milmeister & Weissmann 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66">66</a></li><li><a href="#CITEREFKrennLorünser2023">Krenn & Lorünser 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RRi2EAAAQBAJ&pg=PA8">8</a></li></ul></div></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44">44–45</a></li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136</a></li></ul></div></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKrennLorünser2023">Krenn & Lorünser 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RRi2EAAAQBAJ&pg=PA8">8</a></li><li><a href="#CITEREFMazzolaMilmeisterWeissmann2004">Mazzola, Milmeister & Weissmann 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66">66</a></li></ul></div></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87">87</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li></ul></div></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA25">25</a></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><a href="#CITEREFConfrey1994">Confrey 1994</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4MwZlzgvaGYC&pg=PA308">308</a></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA93">93–94</a></li><li><a href="#CITEREFKay2021">Kay 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44">44–45</a></li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136</a></li></ul></div></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFWheater2015">Wheater 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19">19</a></li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136–137</a></li><li><a href="#CITEREFAchatzAnderson2005">Achatz & Anderson 2005</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YOdtemSmzQQC&pg=PA18">18</a></li></ul></div></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMazzolaMilmeisterWeissmann2004">Mazzola, Milmeister & Weissmann 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66">66</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 179–180</li></ul></div></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101">101–102</a></li></ul></div></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><a href="#CITEREFCavanagh2017">Cavanagh 2017</a>, p. 275</span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101">101–102</a></li></ul></div></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFWheater2015">Wheater 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19">19</a></li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136</a></li></ul></div></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA117">117</a></li><li><a href="#CITEREFWheater2015">Wheater 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19">19</a></li><li><a href="#CITEREFWrightEllemor-CollinsTabor2011">Wright, Ellemor-Collins & Tabor 2011</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136">136–137</a></li></ul></div></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMazzolaMilmeisterWeissmann2004">Mazzola, Milmeister & Weissmann 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66">66</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, pp. 303–304</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 179–180</li></ul></div></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA117">117–118</a></li><li><a href="#CITEREFKay2021">Kay 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA27">27–28</a></li></ul></div></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA120">120</a></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA118">118</a></li><li><a href="#CITEREFKlose2014">Klose 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA105">105</a></li></ul></div></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA121">121–122</a></li><li><a href="#CITEREFRoddaLittle2015">Rodda & Little 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cb_dCgAAQBAJ&pg=PA7">7</a></li></ul></div></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA117">117</a></li><li><a href="#CITEREFMazzolaMilmeisterWeissmann2004">Mazzola, Milmeister & Weissmann 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66">66</a></li></ul></div></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFSallySally_(Jr.)2012">Sally & Sally (Jr.) 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA3">3</a></li><li><a href="#CITEREFKlose2014">Klose 2014</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA107">107–108</a></li></ul></div></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNagel2002">Nagel 2002</a>, pp. 180–181</li><li><a href="#CITEREFGupta2019">Gupta 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vcmcDwAAQBAJ&pg=PA3">3</a></li><li><a href="#CITEREFVaccaroPepiciello2022">Vaccaro & Pepiciello 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tZxBEAAAQBAJ&pg=PA9">9–12</a></li><li><a href="#CITEREFLiebler2018">Liebler 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ozb3DwAAQBAJ&pg=PA36">36</a></li></ul></div></span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFHafstrom2013">Hafstrom 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA95">95</a></li><li><a href="#CITEREFBrentZimmermann2010">Brent & Zimmermann 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-8wuH5AwbwMC&pg=PA1">1</a></li></ul></div></span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKupferman2015">Kupferman 2015</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cd87DQAAQBAJ&pg=PA92">45, 92</a></li><li><a href="#CITEREFUspenskiiSemenov2001">Uspenskii & Semenov 2001</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yID37VIW-t8C&pg=PA113">113</a></li><li><a href="#CITEREFGeary2006">Geary 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796">796</a></li></ul></div></span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFResnickFord2012">Resnick & Ford 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xj-j8pw2HN8C&pg=PA110">110</a></li><li><a href="#CITEREFKleinMoellerDresselDomahs2010">Klein et al. 2010</a>, pp. 67–68</li></ul></div></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFQuinteroRosario2016">Quintero & Rosario 2016</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YAXyCwAAQBAJ&pg=PA74">74</a></li><li><a href="#CITEREFEbbyHulbertBroadhead2020">Ebby, Hulbert & Broadhead 2020</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=shEHEAAAQBAJ&pg=PA24">24–26</a></li></ul></div></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><a href="#CITEREFSperlingStuart1981">Sperling & Stuart 1981</a>, p. 7</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><a href="#CITEREFSperlingStuart1981">Sperling & Stuart 1981</a>, p. 8</span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMa2020">Ma 2020</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2Tr3DwAAQBAJ&pg=PA35">35–36</a></li><li><a href="#CITEREFSperlingStuart1981">Sperling & Stuart 1981</a>, p. 9</li></ul></div></span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><a href="#CITEREFMooneyBriggsHansenMcCullouch2014">Mooney et al. 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_dPgAwAAQBAJ&pg=PT148">148</a></span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKlein2013">Klein 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GYpEAAAAQBAJ&pg=PA249">249</a></li><li><a href="#CITEREFMullerBrunieDinechinJeannerod2018">Muller et al. 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h3ZZDwAAQBAJ&pg=PA539">539</a></li></ul></div></span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text"><a href="#CITEREFDavisGouldingSuggate2017">Davis, Goulding & Suggate 2017</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7T8lDwAAQBAJ&pg=PA11">11–12</a></span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text"><a href="#CITEREFHaylockCockburn2008">Haylock & Cockburn 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hgAr3maZeQUC&pg=PA49">49</a></span> </li> <li id="cite_note-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-79">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPrata2002">Prata 2002</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MsizNs-zVMAC&pg=PA138">138</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA135">135–136</a></li></ul></div></span> </li> <li id="cite_note-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-80">^</a></b></span> <span class="reference-text"><a href="#CITEREFKoepf2021">Koepf 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rOU3EAAAQBAJ&pg=PA49">49</a></span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoodstein2014">Goodstein 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EHjiBQAAQBAJ&pg=PA33">33</a></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCafaroEpicocoPulimeno2018">Cafaro, Epicoco & Pulimeno 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rs51DwAAQBAJ&pg=PA7">7</a></li><li><a href="#CITEREFReilly2009">Reilly 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q33he4hOlKcC&pg=PA75">75</a></li></ul></div></span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCuytPetersenVerdonkWaadeland2008">Cuyt et al. 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182">182</a></li><li><a href="#CITEREFMahajan2010">Mahajan 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VrkZN0T0GaUC&pg=PA66">66–69</a></li><li><a href="#CITEREFLang2002">Lang 2002</a>, pp. 205–206</li></ul></div></span> </li> <li id="cite_note-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-84">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKay2021">Kay 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA57">57</a></li><li><a href="#CITEREFCuytPetersenVerdonkWaadeland2008">Cuyt et al. 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182">182</a></li></ul></div></span> </li> <li id="cite_note-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-85">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li><li><a href="#CITEREFGrigorieva2018">Grigorieva 2018</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mEpjDwAAQBAJ&pg=PR8">viii–ix</a></li><li><a href="#CITEREFPage2003">Page 2003</a>, p. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">15</a></li></ul></div></span> </li> <li id="cite_note-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-86">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPage2003">Page 2003</a>, p. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">34</a></li><li><a href="#CITEREFYan2002">Yan 2002</a>, p. 12</li></ul></div></span> </li> <li id="cite_note-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-87">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPage2003">Page 2003</a>, pp. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">18–19, 34</a></li><li><a href="#CITEREFBukhshtabNechaev2014">Bukhshtab & Nechaev 2014</a></li></ul></div></span> </li> <li id="cite_note-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-88">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPage2003">Page 2003</a>, p. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">34</a></li><li><a href="#CITEREFKaratsuba2014">Karatsuba 2014</a></li></ul></div></span> </li> <li id="cite_note-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-89">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFPage2003">Page 2003</a>, pp. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">34–35</a></li><li><a href="#CITEREFVinogradov2019">Vinogradov 2019</a></li></ul></div></span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text"><a href="#CITEREFKubilyus2018">Kubilyus 2018</a></span> </li> <li id="cite_note-91"><span class="mw-cite-backlink"><b><a href="#cite_ref-91">^</a></b></span> <span class="reference-text"><a href="#CITEREFPomeranceSárközy1995">Pomerance & Sárközy 1995</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ktBP5vUl5gC&pg=PA969">969</a></span> </li> <li id="cite_note-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-92">^</a></b></span> <span class="reference-text"><a href="#CITEREFPomerance2010">Pomerance 2010</a></span> </li> <li id="cite_note-93"><span class="mw-cite-backlink"><b><a href="#cite_ref-93">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFYan2002">Yan 2002</a>, pp. 12, 303–305</li><li><a href="#CITEREFYan2013a">Yan 2013a</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15">15</a></li></ul></div></span> </li> <li id="cite_note-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-94">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li><li><a href="#CITEREFKřížekSomerŠolcová2021">Křížek, Somer & Šolcová 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23">23, 25, 37</a></li></ul></div></span> </li> <li id="cite_note-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-95">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKřížekSomerŠolcová2021">Křížek, Somer & Šolcová 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23">23</a></li><li><a href="#CITEREFRiesel2012">Riesel 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA2">2</a></li></ul></div></span> </li> <li id="cite_note-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-96">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li><li><a href="#CITEREFKřížekSomerŠolcová2021">Křížek, Somer & Šolcová 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA25">25</a></li></ul></div></span> </li> <li id="cite_note-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-97">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBukhshtabNechaev2016">Bukhshtab & Nechaev 2016</a></li><li><a href="#CITEREFKřížekSomerŠolcová2021">Křížek, Somer & Šolcová 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA37">37</a></li></ul></div></span> </li> <li id="cite_note-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-98">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA30">30</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 304</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. x</li><li><a href="#CITEREFHafstrom2013">Hafstrom 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA123">123</a></li><li><a href="#CITEREFCohen2003">Cohen 2003</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=URK2DwAAQBAJ&pg=PA37">37</a></li></ul></div></span> </li> <li id="cite_note-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-99">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA31">31–32</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA347">347</a></li></ul></div></span> </li> <li id="cite_note-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-100">^</a></b></span> <span class="reference-text"><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA32">32–33</a></span> </li> <li id="cite_note-101"><span class="mw-cite-backlink"><b><a href="#cite_ref-101">^</a></b></span> <span class="reference-text"><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA33">33</a></span> </li> <li id="cite_note-102"><span class="mw-cite-backlink"><b><a href="#cite_ref-102">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlose2014">Klose 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA107">107</a></span> </li> <li id="cite_note-103"><span class="mw-cite-backlink"><b><a href="#cite_ref-103">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHoffmanFrankel2018">Hoffman & Frankel 2018</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=F5K3DwAAQBAJ&pg=PA161">161–162</a></li><li><a href="#CITEREFLange2010">Lange 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AtiDhx2bsiMC&pg=PA248">248–249</a></li><li><a href="#CITEREFKlose2014">Klose 2014</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA105">105–107</a></li></ul></div></span> </li> <li id="cite_note-104"><span class="mw-cite-backlink"><b><a href="#cite_ref-104">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCuytPetersenVerdonkWaadeland2008">Cuyt et al. 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182">182</a></li><li><a href="#CITEREFMahajan2010">Mahajan 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VrkZN0T0GaUC&pg=PA66">66–69</a></li></ul></div></span> </li> <li id="cite_note-105"><span class="mw-cite-backlink"><b><a href="#cite_ref-105">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA33">33</a></li><li><a href="#CITEREFIgarashiAltmanFunadaKamiyama2014">Igarashi et al. 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA18">18</a></li></ul></div></span> </li> <li id="cite_note-106"><span class="mw-cite-backlink"><b><a href="#cite_ref-106">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA35">35</a></li><li><a href="#CITEREFBookerBondSparrowSwan2015">Booker et al. 2015</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lTLiBAAAQBAJ&pg=PA308">308–309</a></li></ul></div></span> </li> <li id="cite_note-107"><span class="mw-cite-backlink"><b><a href="#cite_ref-107">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGellertHellwichKästnerKüstner2012">Gellert et al. 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA34">34</a></li><li><a href="#CITEREFIgarashiAltmanFunadaKamiyama2014">Igarashi et al. 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA18">18</a></li></ul></div></span> </li> <li id="cite_note-108"><span class="mw-cite-backlink"><b><a href="#cite_ref-108">^</a></b></span> <span class="reference-text"><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358">358</a></span> </li> <li id="cite_note-109"><span class="mw-cite-backlink"><b><a href="#cite_ref-109">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358">358–359</a></li><li><a href="#CITEREFKudryavtsev2020">Kudryavtsev 2020</a></li><li><a href="#CITEREFRooney2021">Rooney 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34">34</a></li><li><a href="#CITEREFYoung2010">Young 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA994">994–996</a></li><li><a href="#CITEREFFarmer2023">Farmer 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VfOkEAAAQBAJ&pg=PA139">139</a></li></ul></div></span> </li> <li id="cite_note-110"><span class="mw-cite-backlink"><b><a href="#cite_ref-110">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRossi2011">Rossi 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kSwVGbBtel8C&pg=PA101">101</a></li><li><a href="#CITEREFReitano2010">Reitano 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JYX6AQAAQBAJ&pg=PA42">42</a></li><li><a href="#CITEREFBronshteinSemendyayevMusiolMühlig2015">Bronshtein et al. 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5L6BBwAAQBAJ&pg=PA2">2</a></li></ul></div></span> </li> <li id="cite_note-111"><span class="mw-cite-backlink"><b><a href="#cite_ref-111">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358">358–359</a></li><li><a href="#CITEREFKudryavtsev2020">Kudryavtsev 2020</a></li><li><a href="#CITEREFRooney2021">Rooney 2021</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34">34</a></li><li><a href="#CITEREFYoung2010">Young 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA994">994–996</a></li></ul></div></span> </li> <li id="cite_note-112"><span class="mw-cite-backlink"><b><a href="#cite_ref-112">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFWallis2013">Wallis 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ONgRBwAAQBAJ&pg=PA20">20–21</a></li><li><a href="#CITEREFYoung2010">Young 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA996">996–997</a></li><li><a href="#CITEREFYoung2021">Young 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hpVFEAAAQBAJ&pg=RA1-PA4">4–5</a></li></ul></div></span> </li> <li id="cite_note-113"><span class="mw-cite-backlink"><b><a href="#cite_ref-113">^</a></b></span> <span class="reference-text"><a href="#CITEREFKoren2018">Koren 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wUBZDwAAQBAJ&pg=PA71">71</a></span> </li> <li id="cite_note-114"><span class="mw-cite-backlink"><b><a href="#cite_ref-114">^</a></b></span> <span class="reference-text"><a href="#CITEREFDrosg2007">Drosg 2007</a>, pp. 1–5</span> </li> <li id="cite_note-115"><span class="mw-cite-backlink"><b><a href="#cite_ref-115">^</a></b></span> <span class="reference-text"><a href="#CITEREFBohacek2009">Bohacek 2009</a>, pp. 18–19</span> </li> <li id="cite_note-116"><span class="mw-cite-backlink"><b><a href="#cite_ref-116">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHigham2002">Higham 2002</a>, pp. 3–5</li><li><a href="#CITEREFBohacek2009">Bohacek 2009</a>, pp. 8–19</li></ul></div></span> </li> <li id="cite_note-117"><span class="mw-cite-backlink"><b><a href="#cite_ref-117">^</a></b></span> <span class="reference-text"><a href="#CITEREFBohacek2009">Bohacek 2009</a>, pp. 18–19</span> </li> <li id="cite_note-118"><span class="mw-cite-backlink"><b><a href="#cite_ref-118">^</a></b></span> <span class="reference-text"><a href="#CITEREFBohacek2009">Bohacek 2009</a>, pp. 23–30</span> </li> <li id="cite_note-119"><span class="mw-cite-backlink"><b><a href="#cite_ref-119">^</a></b></span> <span class="reference-text"><a href="#CITEREFGriffin1935">Griffin 1935</a></span> </li> <li id="cite_note-120"><span class="mw-cite-backlink"><b><a href="#cite_ref-120">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMooreKearfottCloud2009">Moore, Kearfott & Cloud 2009</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kd8FmmN7sAoC&pg=PA10">10–11, 19</a></li><li><a href="#CITEREFPharrJakobHumphreys2023">Pharr, Jakob & Humphreys 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kUtwEAAAQBAJ&pg=PA1057">1057</a></li></ul></div></span> </li> <li id="cite_note-121"><span class="mw-cite-backlink"><b><a href="#cite_ref-121">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFVaccaroPepiciello2022">Vaccaro & Pepiciello 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tZxBEAAAQBAJ&pg=PA9">9–11</a></li><li><a href="#CITEREFChakravertyRout2022">Chakraverty & Rout 2022</a>, pp. 2–4, 39–40</li></ul></div></span> </li> <li id="cite_note-122"><span class="mw-cite-backlink"><b><a href="#cite_ref-122">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFWallis2013">Wallis 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ONgRBwAAQBAJ&pg=PA20">20</a></li><li><a href="#CITEREFRoedeForestJamshidi2018">Roe, deForest & Jamshidi 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3ppYDwAAQBAJ&pg=PA24">24</a></li></ul></div></span> </li> <li id="cite_note-123"><span class="mw-cite-backlink"><b><a href="#cite_ref-123">^</a></b></span> <span class="reference-text"><a href="#CITEREFLustick1997">Lustick 1997</a></span> </li> <li id="cite_note-124"><span class="mw-cite-backlink"><b><a href="#cite_ref-124">^</a></b></span> <span class="reference-text"><a href="#CITEREFMullerBrisebarreDinechinJeannerod2009">Muller et al. 2009</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=baFvrIOPvncC&pg=PA13">13–16</a></span> </li> <li id="cite_note-125"><span class="mw-cite-backlink"><b><a href="#cite_ref-125">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKoren2018">Koren 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wUBZDwAAQBAJ&pg=PA71">71</a></li><li><a href="#CITEREFMullerBrisebarreDinechinJeannerod2009">Muller et al. 2009</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=baFvrIOPvncC&pg=PA13">13–16</a></li><li><a href="#CITEREFSwartzlander2017">Swartzlander 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VOnyWUUUj04C&pg=SA11-PA19">11.19</a></li></ul></div></span> </li> <li id="cite_note-126"><span class="mw-cite-backlink"><b><a href="#cite_ref-126">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFStewart2022">Stewart 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=twafEAAAQBAJ&pg=PA26">26</a></li><li><a href="#CITEREFMeyer2023">Meyer 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-X-_EAAAQBAJ&pg=PA234">234</a></li></ul></div></span> </li> <li id="cite_note-127"><span class="mw-cite-backlink"><b><a href="#cite_ref-127">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMullerBrisebarreDinechinJeannerod2009">Muller et al. 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=baFvrIOPvncC&pg=PA54">54</a></li><li><a href="#CITEREFBrentZimmermann2010">Brent & Zimmermann 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-8wuH5AwbwMC&pg=PA79">79</a></li><li><a href="#CITEREFCryer2014">Cryer 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_x3pAwAAQBAJ&pg=PA450">450</a></li></ul></div></span> </li> <li id="cite_note-128"><span class="mw-cite-backlink"><b><a href="#cite_ref-128">^</a></b></span> <span class="reference-text"><a href="#CITEREFDuffy2018">Duffy 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BTttDwAAQBAJ&pg=PT1225">1225</a></span> </li> <li id="cite_note-129"><span class="mw-cite-backlink"><b><a href="#cite_ref-129">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131">131</a></li><li><a href="#CITEREFVerschaffelTorbeynsDe_Smedt2011">Verschaffel, Torbeyns & De Smedt 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177">2177</a></li></ul></div></span> </li> <li id="cite_note-130"><span class="mw-cite-backlink"><b><a href="#cite_ref-130">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFEmersonBabtie2014">Emerson & Babtie 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NQ-aBQAAQBAJ&pg=PA147">147</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131">131–132</a></li><li><a href="#CITEREFVerschaffelTorbeynsDe_Smedt2011">Verschaffel, Torbeyns & De Smedt 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177">2177</a></li></ul></div></span> </li> <li id="cite_note-131"><span class="mw-cite-backlink"><b><a href="#cite_ref-131">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131">131</a></li><li><a href="#CITEREFVerschaffelTorbeynsDe_Smedt2011">Verschaffel, Torbeyns & De Smedt 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177">2177</a></li></ul></div></span> </li> <li id="cite_note-132"><span class="mw-cite-backlink"><b><a href="#cite_ref-132">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFDowker2019">Dowker 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GQaQDwAAQBAJ&pg=PA114">114</a></li><li><a href="#CITEREFBerchGearyKoepke2015">Berch, Geary & Koepke 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XS9OBQAAQBAJ&pg=PA124">124</a></li><li><a href="#CITEREFOtis2024">Otis 2024</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=07jiEAAAQBAJ&pg=PA15">15–19</a></li><li><a href="#CITEREFGeary2006">Geary 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796">796</a></li></ul></div></span> </li> <li id="cite_note-133"><span class="mw-cite-backlink"><b><a href="#cite_ref-133">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOtis2024">Otis 2024</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=07jiEAAAQBAJ&pg=PA15">15–19</a></li><li><a href="#CITEREFGeary2006">Geary 2006</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796">796</a></li></ul></div></span> </li> <li id="cite_note-134"><span class="mw-cite-backlink"><b><a href="#cite_ref-134">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, p. 8</li><li><a href="#CITEREFMazumderEbong2023">Mazumder & Ebong 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18">18</a></li></ul></div></span> </li> <li id="cite_note-135"><span class="mw-cite-backlink"><b><a href="#cite_ref-135">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFReynolds2008">Reynolds 2008</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1">1–2</a></li><li><a href="#CITEREFSternbergBen-Zeev2012">Sternberg & Ben-Zeev 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q7F777rDl1AC&pg=PA95">95–96</a></li></ul></div></span> </li> <li id="cite_note-136"><span class="mw-cite-backlink"><b><a href="#cite_ref-136">^</a></b></span> <span class="reference-text"><a href="#CITEREFBuddSangwin2001">Budd & Sangwin 2001</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x8hoAsL_JL4C&pg=PA209">209</a></span> </li> <li id="cite_note-137"><span class="mw-cite-backlink"><b><a href="#cite_ref-137">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKnoblochKomatsuLiu2013">Knobloch, Komatsu & Liu 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JgDFBAAAQBAJ&pg=PA123">123</a></li><li><a href="#CITEREFHodgkin2013">Hodgkin 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nSO5iMujRUYC&pg=PR168">168</a></li><li><a href="#CITEREFHart2011">Hart 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zLPm3xE2qWgC&pg=PA69">69</a></li></ul></div></span> </li> <li id="cite_note-138"><span class="mw-cite-backlink"><b><a href="#cite_ref-138">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHodgkin2013">Hodgkin 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nSO5iMujRUYC&pg=PR168">168</a></li><li><a href="#CITEREFHart2011">Hart 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zLPm3xE2qWgC&pg=PA69">69</a></li></ul></div></span> </li> <li id="cite_note-140"><span class="mw-cite-backlink"><b><a href="#cite_ref-140">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBruderer2021">Bruderer 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gh8SEAAAQBAJ&pg=PA543">543–545, 906–907</a></li><li><a href="#CITEREFKlaf2011">Klaf 2011</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YRNeBAAAQBAJ&pg=PA187">187–188</a></li></ul></div></span> </li> <li id="cite_note-141"><span class="mw-cite-backlink"><b><a href="#cite_ref-141">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFStrathern2012">Strathern 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NJm7Z5iQE24C&pg=PP9">9</a></li><li><a href="#CITEREFLang2015">Lang 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wbiKDwAAQBAJ&pg=PA160">160</a></li></ul></div></span> </li> <li id="cite_note-143"><span class="mw-cite-backlink"><b><a href="#cite_ref-143">^</a></b></span> <span class="reference-text"><a href="#CITEREFCampbell-KellyCroarkenFloodRobson2007">Campbell-Kelly et al. 2007</a>, p. 2</span> </li> <li id="cite_note-144"><span class="mw-cite-backlink"><b><a href="#cite_ref-144">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLockhart2017">Lockhart 2017</a>, pp. 136, 140–146</li><li><a href="#CITEREFO'Regan2012">O'Regan 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QqrItgm351EC&pg=PA24">24–25</a></li></ul></div></span> </li> <li id="cite_note-145"><span class="mw-cite-backlink"><b><a href="#cite_ref-145">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFKhouryLamothe2016">Khoury & Lamothe 2016</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T162DAAAQBAJ&pg=PA2">2</a></li><li><a href="#CITEREFLockhart2017">Lockhart 2017</a>, pp. 147–150</li><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA119">119</a></li></ul></div></span> </li> <li id="cite_note-146"><span class="mw-cite-backlink"><b><a href="#cite_ref-146">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLernerLerner2008">Lerner & Lerner 2008</a>, pp. 2807–2808</li><li><a href="#CITEREFWallis2011">Wallis 2011</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=18W4_LJ5bL0C&pg=PA303">303–304</a></li><li><a href="#CITEREFKaiserGranade2021">Kaiser & Granade 2021</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IxIxEAAAQBAJ&pg=PA283">283–284</a></li></ul></div></span> </li> <li id="cite_note-147"><span class="mw-cite-backlink"><b><a href="#cite_ref-147">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMooreKearfottCloud2009">Moore, Kearfott & Cloud 2009</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kd8FmmN7sAoC&pg=PA10">10–11, 19</a></li><li><a href="#CITEREFPharrJakobHumphreys2023">Pharr, Jakob & Humphreys 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kUtwEAAAQBAJ&pg=PA1057">1057</a></li></ul></div></span> </li> <li id="cite_note-148"><span class="mw-cite-backlink"><b><a href="#cite_ref-148">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLiebler2018">Liebler 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ozb3DwAAQBAJ&pg=PA36">36</a></li><li><a href="#CITEREFAdhamiMeenenMeenenHite2007">Adhami et al. 2007</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9nqkVbFPutYC&pg=PA80">80–82, 98–102</a></li></ul></div></span> </li> <li id="cite_note-149"><span class="mw-cite-backlink"><b><a href="#cite_ref-149">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFShiva2018">Shiva 2018</a>, pp. 3, 14</li><li><a href="#CITEREFGupta2019">Gupta 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vcmcDwAAQBAJ&pg=PA3">3</a></li></ul></div></span> </li> <li id="cite_note-150"><span class="mw-cite-backlink"><b><a href="#cite_ref-150">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. 92–93</span> </li> <li id="cite_note-151"><span class="mw-cite-backlink"><b><a href="#cite_ref-151">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. xviii–xx, xxiv, 137–138</li><li><a href="#CITEREFCaprioAveniMukherjee2022">Caprio, Aveni & Mukherjee 2022</a>, pp. 763–764</li></ul></div></span> </li> <li id="cite_note-152"><span class="mw-cite-backlink"><b><a href="#cite_ref-152">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. 144</li><li><a href="#CITEREFCaprioAveniMukherjee2022">Caprio, Aveni & Mukherjee 2022</a>, pp. 763–764</li><li><a href="#CITEREFSeamanRosslerBurgin2023">Seaman, Rossler & Burgin 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=213PEAAAQBAJ&pg=PA226">226</a></li></ul></div></span> </li> <li id="cite_note-153"><span class="mw-cite-backlink"><b><a href="#cite_ref-153">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFBukhshtabPechaev2020">Bukhshtab & Pechaev 2020</a></li><li><a href="#CITEREFTiles2009">Tiles 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mbn35b2ghgkC&pg=PA243">243</a></li></ul></div></span> </li> <li id="cite_note-155"><span class="mw-cite-backlink"><b><a href="#cite_ref-155">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFFerreiros2013">Ferreiros 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-WL0BwAAQBAJ&pg=PA251">251</a></li><li><a href="#CITEREFOngleyCarey2013">Ongley & Carey 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26">26–27</a></li></ul></div></span> </li> <li id="cite_note-156"><span class="mw-cite-backlink"><b><a href="#cite_ref-156">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFOngleyCarey2013">Ongley & Carey 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26">26–27</a></li><li><a href="#CITEREFXuZhang2022">Xu & Zhang 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s2RXEAAAQBAJ&pg=PA121">121</a></li></ul></div></span> </li> <li id="cite_note-157"><span class="mw-cite-backlink"><b><a href="#cite_ref-157">^</a></b></span> <span class="reference-text"><a href="#CITEREFTaylor2012">Taylor 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8">8</a></span> </li> <li id="cite_note-159"><span class="mw-cite-backlink"><b><a href="#cite_ref-159">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOngleyCarey2013">Ongley & Carey 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26">26–27</a></li><li><a href="#CITEREFTaylor2012">Taylor 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8">8</a></li></ul></div></span> </li> <li id="cite_note-160"><span class="mw-cite-backlink"><b><a href="#cite_ref-160">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBagaria2023">Bagaria 2023</a>, § 3. The Theory of Transfinite Ordinals and Cardinals</li><li><a href="#CITEREFCunningham2016">Cunningham 2016</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fdigDAAAQBAJ&pg=PA83">83–84, 108</a></li></ul></div></span> </li> <li id="cite_note-161"><span class="mw-cite-backlink"><b><a href="#cite_ref-161">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHamiltonLandin2018">Hamilton & Landin 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA133">133</a></li><li><a href="#CITEREFBagaria2023">Bagaria 2023</a>, § 5. Set Theory as the Foundation of Mathematics</li></ul></div></span> </li> <li id="cite_note-162"><span class="mw-cite-backlink"><b><a href="#cite_ref-162">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHamiltonLandin2018">Hamilton & Landin 2018</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA157">157–158</a></li><li><a href="#CITEREFBagaria2023">Bagaria 2023</a>, § 5. Set Theory as the Foundation of Mathematics</li></ul></div></span> </li> <li id="cite_note-163"><span class="mw-cite-backlink"><b><a href="#cite_ref-163">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBagaria2023">Bagaria 2023</a>, § 5. Set Theory as the Foundation of Mathematics</li><li><a href="#CITEREFHamiltonLandin2018">Hamilton & Landin 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA252">252</a></li></ul></div></span> </li> <li id="cite_note-164"><span class="mw-cite-backlink"><b><a href="#cite_ref-164">^</a></b></span> <span class="reference-text"><a href="#CITEREFCunningham2016">Cunningham 2016</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fdigDAAAQBAJ&pg=PA95">95–96</a></span> </li> <li id="cite_note-165"><span class="mw-cite-backlink"><b><a href="#cite_ref-165">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2">2–3</a></li><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 1, 6, 8, 10</li><li><a href="#CITEREFThiamRochon2019">Thiam & Rochon 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164">164</a></li></ul></div></span> </li> <li id="cite_note-166"><span class="mw-cite-backlink"><b><a href="#cite_ref-166">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA3">3</a></li><li><a href="#CITEREFPonticorvoSchmbriMiglino2019">Ponticorvo, Schmbri & Miglino 2019</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zSiXDwAAQBAJ&pg=PA33">33</a></li></ul></div></span> </li> <li id="cite_note-167"><span class="mw-cite-backlink"><b><a href="#cite_ref-167">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA4">4–6</a></li><li><a href="#CITEREFAngLam2004">Ang & Lam 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GxDJCgAAQBAJ&pg=PA170">170</a></li></ul></div></span> </li> <li id="cite_note-168"><span class="mw-cite-backlink"><b><a href="#cite_ref-168">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA5">5–7, 9–11</a></li><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 10–15</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, p. 178</li><li><a href="#CITEREFHindry2011">Hindry 2011</a>, p. ix</li></ul></div></span> </li> <li id="cite_note-169"><span class="mw-cite-backlink"><b><a href="#cite_ref-169">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA6">6–7, 9</a></li><li><a href="#CITEREFOre1948">Ore 1948</a>, pp. 16–18</li><li><a href="#CITEREFITL_Education_Solutions_Limited2011">ITL Education Solutions Limited 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28">28</a></li></ul></div></span> </li> <li id="cite_note-170"><span class="mw-cite-backlink"><b><a href="#cite_ref-170">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOre1948">Ore 1948</a>, p. 15</li><li><a href="#CITEREFYadin2016">Yadin 2016</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KzeLDQAAQBAJ&pg=PT24">24</a></li></ul></div></span> </li> <li id="cite_note-171"><span class="mw-cite-backlink"><b><a href="#cite_ref-171">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA3">4–5</a></li><li><a href="#CITEREFBrown2010">Brown 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TzrNgAsJY1MC&pg=PA184">184</a></li></ul></div></span> </li> <li id="cite_note-172"><span class="mw-cite-backlink"><b><a href="#cite_ref-172">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA15">15</a></li><li><a href="#CITEREFBrown2010">Brown 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TzrNgAsJY1MC&pg=PA184">184</a></li><li><a href="#CITEREFRomanowski2008">Romanowski 2008</a>, p. 303</li><li><a href="#CITEREFNagel2002">Nagel 2002</a>, p. 178</li></ul></div></span> </li> <li id="cite_note-173"><span class="mw-cite-backlink"><b><a href="#cite_ref-173">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA15">15</a></li><li><a href="#CITEREFMaddenAubrey2017">Madden & Aubrey 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6EkzDwAAQBAJ&pg=PP18">xvii</a></li></ul></div></span> </li> <li id="cite_note-174"><span class="mw-cite-backlink"><b><a href="#cite_ref-174">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA31">31</a></li><li><a href="#CITEREFPayne2017">Payne 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qMU2DwAAQBAJ&pg=PA202">202</a></li></ul></div></span> </li> <li id="cite_note-175"><span class="mw-cite-backlink"><b><a href="#cite_ref-175">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA20">20–21</a></li><li><a href="#CITEREFBloch2011">Bloch 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vXw_AAAAQBAJ&pg=PA52">52</a></li></ul></div></span> </li> <li id="cite_note-176"><span class="mw-cite-backlink"><b><a href="#cite_ref-176">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA16">16</a></li><li><a href="#CITEREFLützen2023">Lützen 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=joikEAAAQBAJ&pg=PA19">19</a></li></ul></div></span> </li> <li id="cite_note-177"><span class="mw-cite-backlink"><b><a href="#cite_ref-177">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA29">29–31</a></li><li><a href="#CITEREFKlein2013a">Klein 2013a</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EkPDAgAAQBAJ&pg=PT12">12</a></li></ul></div></span> </li> <li id="cite_note-178"><span class="mw-cite-backlink"><b><a href="#cite_ref-178">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA36">36–37</a></li><li><a href="#CITEREFBradley2006">Bradley 2006</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EIdtVPeD7GcC&pg=PA82">82–83</a></li><li><a href="#CITEREFConradieGoranko2015">Conradie & Goranko 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268">268</a></li></ul></div></span> </li> <li id="cite_note-179"><span class="mw-cite-backlink"><b><a href="#cite_ref-179">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA35">35–36</a></li><li><a href="#CITEREFCai2023">Cai 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DFLNEAAAQBAJ&pg=PA110">110</a></li></ul></div></span> </li> <li id="cite_note-180"><span class="mw-cite-backlink"><b><a href="#cite_ref-180">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA37">37, 40</a></li><li><a href="#CITEREFBradley2006">Bradley 2006</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EIdtVPeD7GcC&pg=PA82">82–83</a></li><li><a href="#CITEREFConradieGoranko2015">Conradie & Goranko 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268">268</a></li></ul></div></span> </li> <li id="cite_note-181"><span class="mw-cite-backlink"><b><a href="#cite_ref-181">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHuaFeng2020">Hua & Feng 2020</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6_sOEAAAQBAJ&pg=PA119">119–120</a></li><li><a href="#CITEREFChemlaKellerProust2023">Chemla, Keller & Proust 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=afKkEAAAQBAJ&pg=PA47">47</a></li></ul></div></span> </li> <li id="cite_note-182"><span class="mw-cite-backlink"><b><a href="#cite_ref-182">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA13">13, 34–35</a></li><li><a href="#CITEREFConradieGoranko2015">Conradie & Goranko 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268">268</a></li></ul></div></span> </li> <li id="cite_note-183"><span class="mw-cite-backlink"><b><a href="#cite_ref-183">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA13">13, 34</a></li><li><a href="#CITEREFConradieGoranko2015">Conradie & Goranko 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268">268</a></li></ul></div></span> </li> <li id="cite_note-184"><span class="mw-cite-backlink"><b><a href="#cite_ref-184">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA38">38, 43–46</a></li><li><a href="#CITEREFConradieGoranko2015">Conradie & Goranko 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268">268</a></li></ul></div></span> </li> <li id="cite_note-185"><span class="mw-cite-backlink"><b><a href="#cite_ref-185">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA56">56</a></li><li><a href="#CITEREFOakes2020">Oakes 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RtyPEAAAQBAJ&pg=PA330">330</a></li></ul></div></span> </li> <li id="cite_note-186"><span class="mw-cite-backlink"><b><a href="#cite_ref-186">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA55">55</a></li><li><a href="#CITEREFWedell2015">Wedell 2015</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Uh5pCgAAQBAJ&pg=PA1235">1235–1236</a></li></ul></div></span> </li> <li id="cite_note-187"><span class="mw-cite-backlink"><b><a href="#cite_ref-187">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA62">62</a></li><li><a href="#CITEREFLützen2023">Lützen 2023</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=joikEAAAQBAJ&pg=PA124">124</a></li></ul></div></span> </li> <li id="cite_note-188"><span class="mw-cite-backlink"><b><a href="#cite_ref-188">^</a></b></span> <span class="reference-text"><a href="#CITEREFVullo2020">Vullo 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g67QDwAAQBAJ&pg=PA140">140</a></span> </li> <li id="cite_note-189"><span class="mw-cite-backlink"><b><a href="#cite_ref-189">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCignoniCossu2016">Cignoni & Cossu 2016</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LridDQAAQBAJ&pg=PA103">103</a></li><li><a href="#CITEREFKoetsier2018">Koetsier 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9jF7DwAAQBAJ&pg=PA255">255</a></li><li><a href="#CITEREFIgarashiAltmanFunadaKamiyama2014">Igarashi et al. 2014</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA87">87–89</a></li></ul></div></span> </li> <li id="cite_note-190"><span class="mw-cite-backlink"><b><a href="#cite_ref-190">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA77">77</a></li><li><a href="#CITEREFErikssonEstepJohnson2013">Eriksson, Estep & Johnson 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FD8mBQAAQBAJ&pg=PA474">474</a></li></ul></div></span> </li> <li id="cite_note-191"><span class="mw-cite-backlink"><b><a href="#cite_ref-191">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA68">68–72</a></li><li><a href="#CITEREFWeil2009">Weil 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ar7gBwAAQBAJ&pg=PR9">ix</a></li></ul></div></span> </li> <li id="cite_note-192"><span class="mw-cite-backlink"><b><a href="#cite_ref-192">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2">2, 88, 95–97</a></li><li><a href="#CITEREFWang1997">Wang 1997</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pckvCy6L_ocC&pg=PA334">334</a></li></ul></div></span> </li> <li id="cite_note-193"><span class="mw-cite-backlink"><b><a href="#cite_ref-193">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFBurgin2022">Burgin 2022</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA119">119, 124</a></li><li><a href="#CITEREFCurley2011">Curley 2011</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EdGbAAAAQBAJ&pg=PA5">5, 19</a></li><li><a href="#CITEREFIgarashiAltmanFunadaKamiyama2014">Igarashi et al. 2014</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA149">149</a></li></ul></div></span> </li> <li id="cite_note-194"><span class="mw-cite-backlink"><b><a href="#cite_ref-194">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNCTM_Staff">NCTM Staff</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics, p. 44, p. 130</li><li><a href="#CITEREFOdomBarbarinWasik2009">Odom, Barbarin & Wasik 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MIuyhc8W-A8C&pg=PA589">589</a></li></ul></div></span> </li> <li id="cite_note-195"><span class="mw-cite-backlink"><b><a href="#cite_ref-195">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLaskiJor’danDaoustMurray2015">Laski et al. 2015</a>, pp. 1–3</li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA59">59, 90–91, 93–94, 106–108</a></li><li><a href="#CITEREFNurnberger-Haag2017">Nurnberger-Haag 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Uls6DwAAQBAJ&pg=PA215">215</a></li></ul></div></span> </li> <li id="cite_note-196"><span class="mw-cite-backlink"><b><a href="#cite_ref-196">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNCTM_Staff">NCTM Staff</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA208">208, 304, 340, 362</a></li></ul></div></span> </li> <li id="cite_note-197"><span class="mw-cite-backlink"><b><a href="#cite_ref-197">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFNCTM_Staff">NCTM Staff</a></li><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics</li><li><a href="#CITEREFCarraherSchliemann2015">Carraher & Schliemann 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lpFGCgAAQBAJ&pg=PA197">197</a></li><li><a href="#CITEREFRuthven2012">Ruthven 2012</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RjZzAgAAQBAJ&pg=PA435">435, 443–444</a></li></ul></div></span> </li> <li id="cite_note-198"><span class="mw-cite-backlink"><b><a href="#cite_ref-198">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFDe_CruzNethSchlimm2010">De Cruz, Neth & Schlimm 2010</a>, pp. <a rel="nofollow" class="external text" href="https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content">59–60</a></li><li><a href="#CITEREFGriceKempMortonGrace2023">Grice et al. 2023</a>, Abstract</li></ul></div></span> </li> <li id="cite_note-199"><span class="mw-cite-backlink"><b><a href="#cite_ref-199">^</a></b></span> <span class="reference-text"><a href="#CITEREFDe_CruzNethSchlimm2010">De Cruz, Neth & Schlimm 2010</a>, pp. <a rel="nofollow" class="external text" href="https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content">60–62</a></span> </li> <li id="cite_note-200"><span class="mw-cite-backlink"><b><a href="#cite_ref-200">^</a></b></span> <span class="reference-text"><a href="#CITEREFDe_CruzNethSchlimm2010">De Cruz, Neth & Schlimm 2010</a>, p. <a rel="nofollow" class="external text" href="https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content">63</a></span> </li> <li id="cite_note-201"><span class="mw-cite-backlink"><b><a href="#cite_ref-201">^</a></b></span> <span class="reference-text"><a href="#CITEREFGriceKempMortonGrace2023">Grice et al. 2023</a>, Abstract</span> </li> <li id="cite_note-202"><span class="mw-cite-backlink"><b><a href="#cite_ref-202">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFVictoria_Department_of_Education_Staff2023">Victoria Department of Education Staff 2023</a></li><li><a href="#CITEREFAskew2010">Askew 2010</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yqbSaXf0RKwC&pg=PA33">33–34</a></li><li><a href="#CITEREFDreeben-Irimia2010">Dreeben-Irimia 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L3gqsXsoiiAC&pg=PA102">102</a></li></ul></div></span> </li> <li id="cite_note-203"><span class="mw-cite-backlink"><b><a href="#cite_ref-203">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFVictoria_Department_of_Education_Staff2023">Victoria Department of Education Staff 2023</a></li><li><a href="#CITEREFBarnesRiceHanoch2017">Barnes, Rice & Hanoch 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yzMlDwAAQBAJ&pg=PA196">196</a></li><li><a href="#CITEREFGerardiGoetteMeier2013">Gerardi, Goette & Meier 2013</a>, pp. 11267–11268</li><li><a href="#CITEREFJackson2008">Jackson 2008</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tLuPAgAAQBAJ&pg=PA152">152</a></li></ul></div></span> </li> <li id="cite_note-204"><span class="mw-cite-backlink"><b><a href="#cite_ref-204">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHofweber2016">Hofweber 2016</a>, pp. 153–154, 162–163</li><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFSierpinskaLerman1996">Sierpinska & Lerman 1996</a>, p. 827</li></ul></div></span> </li> <li id="cite_note-205"><span class="mw-cite-backlink"><b><a href="#cite_ref-205">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFHorsten2023">Horsten 2023</a>, § 3. Platonism</li></ul></div></span> </li> <li id="cite_note-FOOTNOTEColyvan2023Lead_Section-206"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEColyvan2023Lead_Section_206-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFColyvan2023">Colyvan 2023</a>, Lead Section.</span> </li> <li id="cite_note-208"><span class="mw-cite-backlink"><b><a href="#cite_ref-208">^</a></b></span> <span class="reference-text"><a href="#CITEREFHorsten2023">Horsten 2023</a>, § 2.2 Intuitionism</span> </li> <li id="cite_note-209"><span class="mw-cite-backlink"><b><a href="#cite_ref-209">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHorsten2023">Horsten 2023</a>, § 2.1 Logicism</li><li><a href="#CITEREFHofweber2016">Hofweber 2016</a>, pp. 174–175</li></ul></div></span> </li> <li id="cite_note-210"><span class="mw-cite-backlink"><b><a href="#cite_ref-210">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeir2022">Weir 2022</a>, Lead Section</span> </li> <li id="cite_note-211"><span class="mw-cite-backlink"><b><a href="#cite_ref-211">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFSierpinskaLerman1996">Sierpinska & Lerman 1996</a>, p. 830</li></ul></div></span> </li> <li id="cite_note-212"><span class="mw-cite-backlink"><b><a href="#cite_ref-212">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOliver2005">Oliver 2005</a>, p. 58</li><li><a href="#CITEREFSierpinskaLerman1996">Sierpinska & Lerman 1996</a>, pp. 827–876</li></ul></div></span> </li> <li id="cite_note-213"><span class="mw-cite-backlink"><b><a href="#cite_ref-213">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFHorsten2023">Horsten 2023</a>, § 3.2 Naturalism and Indispensability</li><li><a href="#CITEREFSierpinskaLerman1996">Sierpinska & Lerman 1996</a>, p. 830</li></ul></div></span> </li> <li id="cite_note-214"><span class="mw-cite-backlink"><b><a href="#cite_ref-214">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFLockhart2017">Lockhart 2017</a>, pp. 1–2</li><li><a href="#CITEREFBird2021">Bird 2021</a>, p. 3</li><li><a href="#CITEREFAubrey1999">Aubrey 1999</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_NhpY_VPdCsC&pg=PA49">49</a></li></ul></div></span> </li> <li id="cite_note-215"><span class="mw-cite-backlink"><b><a href="#cite_ref-215">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFOmondi2020">Omondi 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_m7NDwAAQBAJ&pg=PR8">viii</a></li><li><a href="#CITEREFPaarPelzl2009">Paar & Pelzl 2009</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f24wFELSzkoC&pg=PA13">13</a></li></ul></div></span> </li> <li id="cite_note-216"><span class="mw-cite-backlink"><b><a href="#cite_ref-216">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA17">17</a></li><li><a href="#CITEREFKleiner2012">Kleiner 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7NcEIteAfyYC&pg=PA255">255</a></li><li><a href="#CITEREFMarcusMcEvoy2016">Marcus & McEvoy 2016</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x29VCwAAQBAJ&pg=PA285">285</a></li><li><a href="#CITEREFMonahan2012">Monahan 2012</a></li></ul></div></span> </li> <li id="cite_note-217"><span class="mw-cite-backlink"><b><a href="#cite_ref-217">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFGallistelGelman2005">Gallistel & Gelman 2005</a>, pp. 559–560</li><li><a href="#CITEREFAli_RahmanShahrillAbbasTan2017">Ali Rahman et al. 2017</a>, pp. 373–374</li><li><a href="#CITEREFLiSchoenfeld2019">Li & Schoenfeld 2019</a>, Abstract, Introducation</li><li><a href="#CITEREFAsano2013">Asano 2013</a>, pp. xiii–xv</li></ul></div></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=28" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAchatzAnderson2005" class="citation book cs1">Achatz, Thomas; Anderson, John G. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YOdtemSmzQQC&pg=PA18"><i>Technical Shop Mathematics</i></a>. Industrial Press Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8311-3086-2" title="Special:BookSources/978-0-8311-3086-2"><bdi>978-0-8311-3086-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Technical+Shop+Mathematics&rft.pub=Industrial+Press+Inc.&rft.date=2005&rft.isbn=978-0-8311-3086-2&rft.aulast=Achatz&rft.aufirst=Thomas&rft.au=Anderson%2C+John+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYOdtemSmzQQC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdamowicz1994" class="citation book cs1">Adamowicz, Zofia (1994). "The Power of Exponentiation in Arithmetic". In Joseph, Anthony; Mignot, Fulbert; Murat, François; Prum, Bernard; Rentschler, Rudolf (eds.). <i>First European Congress of Mathematics: Paris, July 6-10, 1992 Volume I Invited Lectures (Part 1)</i>. Birkhäuser. pp. <span class="nowrap">299–</span>320. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0348-9110-3_9">10.1007/978-3-0348-9110-3_9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-9110-3" title="Special:BookSources/978-3-0348-9110-3"><bdi>978-3-0348-9110-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Power+of+Exponentiation+in+Arithmetic&rft.btitle=First+European+Congress+of+Mathematics%3A+Paris%2C+July+6-10%2C+1992+Volume+I+Invited+Lectures+%28Part+1%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E299-%3C%2Fspan%3E320&rft.pub=Birkh%C3%A4user&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2F978-3-0348-9110-3_9&rft.isbn=978-3-0348-9110-3&rft.aulast=Adamowicz&rft.aufirst=Zofia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdhamiMeenenMeenenHite2007" class="citation book cs1">Adhami, Reza; Meenen, Peter M.; Meenen, Peter; Hite, Denis (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9nqkVbFPutYC&pg=PA77"><i>Fundamental Concepts in Electrical and Computer Engineering with Practical Design Problems</i></a>. Universal-Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58112-971-7" title="Special:BookSources/978-1-58112-971-7"><bdi>978-1-58112-971-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Concepts+in+Electrical+and+Computer+Engineering+with+Practical+Design+Problems&rft.pub=Universal-Publishers&rft.date=2007&rft.isbn=978-1-58112-971-7&rft.aulast=Adhami&rft.aufirst=Reza&rft.au=Meenen%2C+Peter+M.&rft.au=Meenen%2C+Peter&rft.au=Hite%2C+Denis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9nqkVbFPutYC%26pg%3DPA77&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAli_RahmanShahrillAbbasTan2017" class="citation journal cs1">Ali Rahman, Ernna Sukinnah; Shahrill, Masitah; Abbas, Nor Arifahwati; Tan, Abby (2017). "Developing Students' Mathematical Skills Involving Order of Operations". <i>International Journal of Research in Education and Science</i>: 373. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.21890%2Fijres.327896">10.21890/ijres.327896</a> (inactive 12 December 2024).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Research+in+Education+and+Science&rft.atitle=Developing+Students%27+Mathematical+Skills+Involving+Order+of+Operations&rft.pages=373&rft.date=2017&rft_id=info%3Adoi%2F10.21890%2Fijres.327896&rft.aulast=Ali+Rahman&rft.aufirst=Ernna+Sukinnah&rft.au=Shahrill%2C+Masitah&rft.au=Abbas%2C+Nor+Arifahwati&rft.au=Tan%2C+Abby&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of December 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_December_2024" title="Category:CS1 maint: DOI inactive as of December 2024">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAngLam2004" class="citation book cs1">Ang, Tian Se; Lam, Lay Yong (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GxDJCgAAQBAJ&pg=PA170"><i>Fleeting Footsteps: Tracing The Conception Of Arithmetic And Algebra In Ancient China</i></a> (Revised ed.). World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4483-60-5" title="Special:BookSources/978-981-4483-60-5"><bdi>978-981-4483-60-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fleeting+Footsteps%3A+Tracing+The+Conception+Of+Arithmetic+And+Algebra+In+Ancient+China&rft.edition=Revised&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-4483-60-5&rft.aulast=Ang&rft.aufirst=Tian+Se&rft.au=Lam%2C+Lay+Yong&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGxDJCgAAQBAJ%26pg%3DPA170&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAsano2013" class="citation book cs1">Asano, Akihito (2013). <i>An Introduction to Mathematics for Economics</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-00760-4" title="Special:BookSources/978-1-107-00760-4"><bdi>978-1-107-00760-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mathematics+for+Economics&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-107-00760-4&rft.aulast=Asano&rft.aufirst=Akihito&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAskew2010" class="citation book cs1">Askew, Mike (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yqbSaXf0RKwC&pg=PA33">"It Ain't (Just) What You Do: Effective Teachers of Numeracy"</a>. In Ian, Thompson (ed.). <i>Issues In Teaching Numeracy In Primary Schools</i>. McGraw-Hill Education (UK). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-335-24153-8" title="Special:BookSources/978-0-335-24153-8"><bdi>978-0-335-24153-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=It+Ain%27t+%28Just%29+What+You+Do%3A+Effective+Teachers+of+Numeracy&rft.btitle=Issues+In+Teaching+Numeracy+In+Primary+Schools&rft.pub=McGraw-Hill+Education+%28UK%29&rft.date=2010&rft.isbn=978-0-335-24153-8&rft.aulast=Askew&rft.aufirst=Mike&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyqbSaXf0RKwC%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAubrey1999" class="citation book cs1">Aubrey, Carol (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_NhpY_VPdCsC&pg=PA49"><i>A Developmental Approach to Early Numeracy: Helping to Raise Children's Achievements and Deal with Difficulties in Learning</i></a>. A&C Black. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4411-9164-9" title="Special:BookSources/978-1-4411-9164-9"><bdi>978-1-4411-9164-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Developmental+Approach+to+Early+Numeracy%3A+Helping+to+Raise+Children%27s+Achievements+and+Deal+with+Difficulties+in+Learning&rft.pub=A%26C+Black&rft.date=1999&rft.isbn=978-1-4411-9164-9&rft.aulast=Aubrey&rft.aufirst=Carol&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_NhpY_VPdCsC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBagaria2023" class="citation web cs1">Bagaria, Joan (2023). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/set-theory/">"Set Theory"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. Metaphysics Research Lab, Stanford University<span class="reference-accessdate">. Retrieved <span class="nowrap">19 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Set+Theory&rft.date=2023&rft.aulast=Bagaria&rft.aufirst=Joan&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fset-theory%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarnesRiceHanoch2017" class="citation book cs1">Barnes, Andrew J.; Rice, Thomase; Hanoch, Yaniv (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yzMlDwAAQBAJ&pg=PA196">"Using Behavioral Economics to Improve People's Decisions About Purchasing Health Insurance"</a>. In Hanoch, Yaniv; Barnes, Andrew; Rice, Thomas (eds.). <i>Behavioral Economics and Healthy Behaviors: Key Concepts and Current Research</i>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-317-26952-6" title="Special:BookSources/978-1-317-26952-6"><bdi>978-1-317-26952-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Using+Behavioral+Economics+to+Improve+People%27s+Decisions+About+Purchasing+Health+Insurance&rft.btitle=Behavioral+Economics+and+Healthy+Behaviors%3A+Key+Concepts+and+Current+Research&rft.pub=Taylor+%26+Francis&rft.date=2017&rft.isbn=978-1-317-26952-6&rft.aulast=Barnes&rft.aufirst=Andrew+J.&rft.au=Rice%2C+Thomase&rft.au=Hanoch%2C+Yaniv&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyzMlDwAAQBAJ%26pg%3DPA196&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBerchGearyKoepke2015" class="citation book cs1">Berch, Daniel B.; Geary, David C.; Koepke, Kathleen Mann (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XS9OBQAAQBAJ&pg=PA124"><i>Development of Mathematical Cognition: Neural Substrates and Genetic Influences</i></a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-801909-2" title="Special:BookSources/978-0-12-801909-2"><bdi>978-0-12-801909-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Development+of+Mathematical+Cognition%3A+Neural+Substrates+and+Genetic+Influences&rft.pub=Academic+Press&rft.date=2015&rft.isbn=978-0-12-801909-2&rft.aulast=Berch&rft.aufirst=Daniel+B.&rft.au=Geary%2C+David+C.&rft.au=Koepke%2C+Kathleen+Mann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXS9OBQAAQBAJ%26pg%3DPA124&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBird2021" class="citation book cs1">Bird, John (2021). <i>Bird's Engineering Mathematics</i>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-367-64378-2" title="Special:BookSources/978-0-367-64378-2"><bdi>978-0-367-64378-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bird%27s+Engineering+Mathematics&rft.pub=Taylor+%26+Francis&rft.date=2021&rft.isbn=978-0-367-64378-2&rft.aulast=Bird&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBloch2011" class="citation book cs1">Bloch, Ethan D. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vXw_AAAAQBAJ&pg=PA52"><i>The Real Numbers and Real Analysis</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72177-4" title="Special:BookSources/978-0-387-72177-4"><bdi>978-0-387-72177-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Real+Numbers+and+Real+Analysis&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-0-387-72177-4&rft.aulast=Bloch&rft.aufirst=Ethan+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvXw_AAAAQBAJ%26pg%3DPA52&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBohacek2009" class="citation web cs1">Bohacek, Peter (2009). <a rel="nofollow" class="external text" href="https://serc.carleton.edu/quantskills/teaching_methods/uncertainty/examples/example1.html">"Introduction to Measurement"</a>. <i>Teaching Quantitative Skills in the Geosciences</i>. Carleton College Science Education Resource Center; American Association of Physics Teachers<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Teaching+Quantitative+Skills+in+the+Geosciences&rft.atitle=Introduction+to+Measurement&rft.date=2009&rft.aulast=Bohacek&rft.aufirst=Peter&rft_id=https%3A%2F%2Fserc.carleton.edu%2Fquantskills%2Fteaching_methods%2Funcertainty%2Fexamples%2Fexample1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBookerBondSparrowSwan2015" class="citation book cs1">Booker, George; Bond, Denise; Sparrow, Len; Swan, Paul (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lTLiBAAAQBAJ&pg=PA308"><i>Teaching Primary Mathematics</i></a>. Pearson Higher Education AU. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4860-0488-1" title="Special:BookSources/978-1-4860-0488-1"><bdi>978-1-4860-0488-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teaching+Primary+Mathematics&rft.pub=Pearson+Higher+Education+AU&rft.date=2015&rft.isbn=978-1-4860-0488-1&rft.aulast=Booker&rft.aufirst=George&rft.au=Bond%2C+Denise&rft.au=Sparrow%2C+Len&rft.au=Swan%2C+Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlTLiBAAAQBAJ%26pg%3DPA308&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBradley2006" class="citation book cs1">Bradley, Michael J. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EIdtVPeD7GcC&pg=PA82"><i>The Birth of Mathematics: Ancient Times To 1300</i></a>. Infobase Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7910-9723-6" title="Special:BookSources/978-0-7910-9723-6"><bdi>978-0-7910-9723-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Birth+of+Mathematics%3A+Ancient+Times+To+1300&rft.pub=Infobase+Publishing&rft.date=2006&rft.isbn=978-0-7910-9723-6&rft.aulast=Bradley&rft.aufirst=Michael+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEIdtVPeD7GcC%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrentZimmermann2010" class="citation book cs1">Brent, Richard P.; Zimmermann, Paul (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-8wuH5AwbwMC"><i>Modern Computer Arithmetic</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-49228-7" title="Special:BookSources/978-1-139-49228-7"><bdi>978-1-139-49228-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Computer+Arithmetic&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-1-139-49228-7&rft.aulast=Brent&rft.aufirst=Richard+P.&rft.au=Zimmermann%2C+Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-8wuH5AwbwMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBronshteinSemendyayevMusiolMühlig2015" class="citation book cs1">Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5L6BBwAAQBAJ&pg=PA2"><i>Handbook of Mathematics</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-46221-8" title="Special:BookSources/978-3-662-46221-8"><bdi>978-3-662-46221-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematics&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-662-46221-8&rft.aulast=Bronshtein&rft.aufirst=I.+N.&rft.au=Semendyayev%2C+K.+A.&rft.au=Musiol%2C+Gerhard&rft.au=M%C3%BChlig%2C+Heiner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5L6BBwAAQBAJ%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrown2010" class="citation book cs1">Brown, David (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TzrNgAsJY1MC&pg=PA184">"The Measurement of Time and Distance in the Heavens Above Mesopotamia, with Brief Reference Made to Other Ancient Astral Science"</a>. In Morley, Iain; Renfrew, Colin (eds.). <i>The Archaeology of Measurement: Comprehending Heaven, Earth and Time in Ancient Societies</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-11990-0" title="Special:BookSources/978-0-521-11990-0"><bdi>978-0-521-11990-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Measurement+of+Time+and+Distance+in+the+Heavens+Above+Mesopotamia%2C+with+Brief+Reference+Made+to+Other+Ancient+Astral+Science&rft.btitle=The+Archaeology+of+Measurement%3A+Comprehending+Heaven%2C+Earth+and+Time+in+Ancient+Societies&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-11990-0&rft.aulast=Brown&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTzrNgAsJY1MC%26pg%3DPA184&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBruderer2021" class="citation book cs1">Bruderer, Herbert (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gh8SEAAAQBAJ&pg=PA543"><i>Milestones in Analog and Digital Computing</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-40974-6" title="Special:BookSources/978-3-030-40974-6"><bdi>978-3-030-40974-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Milestones+in+Analog+and+Digital+Computing&rft.pub=Springer+Nature&rft.date=2021&rft.isbn=978-3-030-40974-6&rft.aulast=Bruderer&rft.aufirst=Herbert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGh8SEAAAQBAJ%26pg%3DPA543&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBuddSangwin2001" class="citation book cs1">Budd, Christopher J.; Sangwin, Christopher (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x8hoAsL_JL4C&pg=PA209"><i>Mathematics Galore!: Masterclasses, Workshops and Team Projects in Mathematics and Its Applications</i></a>. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850770-3" title="Special:BookSources/978-0-19-850770-3"><bdi>978-0-19-850770-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Galore%21%3A+Masterclasses%2C+Workshops+and+Team+Projects+in+Mathematics+and+Its+Applications&rft.pub=OUP+Oxford&rft.date=2001&rft.isbn=978-0-19-850770-3&rft.aulast=Budd&rft.aufirst=Christopher+J.&rft.au=Sangwin%2C+Christopher&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx8hoAsL_JL4C%26pg%3DPA209&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBukhshtabNechaev2014" class="citation web cs1">Bukhshtab, A. A.; Nechaev, V. I. (2014). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Elementary_number_theory">"Elementary Number Theory"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Elementary+Number+Theory&rft.date=2014&rft.aulast=Bukhshtab&rft.aufirst=A.+A.&rft.au=Nechaev%2C+V.+I.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FElementary_number_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBukhshtabNechaev2016" class="citation web cs1">Bukhshtab, A. A.; Nechaev, V. I. (2016). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Natural_number">"Natural Number"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Natural+Number&rft.date=2016&rft.aulast=Bukhshtab&rft.aufirst=A.+A.&rft.au=Nechaev%2C+V.+I.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FNatural_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBukhshtabPechaev2020" class="citation web cs1">Bukhshtab, A. A.; Pechaev, V. I. (2020). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Arithmetic">"Arithmetic"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Arithmetic&rft.date=2020&rft.aulast=Bukhshtab&rft.aufirst=A.+A.&rft.au=Pechaev%2C+V.+I.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FArithmetic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBurgin2022" class="citation book cs1">Burgin, Mark (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA25"><i>Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-3685-3" title="Special:BookSources/978-981-12-3685-3"><bdi>978-981-12-3685-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trilogy+Of+Numbers+And+Arithmetic+-+Book+1%3A+History+Of+Numbers+And+Arithmetic%3A+An+Information+Perspective&rft.pub=World+Scientific&rft.date=2022&rft.isbn=978-981-12-3685-3&rft.aulast=Burgin&rft.aufirst=Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrWF2EAAAQBAJ%26pg%3DPA25&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCafaroEpicocoPulimeno2018" class="citation book cs1">Cafaro, Massimo; Epicoco, Italo; Pulimeno, Marco (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rs51DwAAQBAJ&pg=PA7">"Techniques for Designing Bioinformatics Algorithms"</a>. <i>Encyclopedia of Bioinformatics and Computational Biology: ABC of Bioinformatics</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-811432-2" title="Special:BookSources/978-0-12-811432-2"><bdi>978-0-12-811432-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Techniques+for+Designing+Bioinformatics+Algorithms&rft.btitle=Encyclopedia+of+Bioinformatics+and+Computational+Biology%3A+ABC+of+Bioinformatics&rft.pub=Elsevier&rft.date=2018&rft.isbn=978-0-12-811432-2&rft.aulast=Cafaro&rft.aufirst=Massimo&rft.au=Epicoco%2C+Italo&rft.au=Pulimeno%2C+Marco&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Drs51DwAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCai2023" class="citation book cs1">Cai, Tianxin (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DFLNEAAAQBAJ&pg=PA110"><i>A Brief History of Mathematics: A Promenade Through the Civilizations of Our World</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-26841-0" title="Special:BookSources/978-3-031-26841-0"><bdi>978-3-031-26841-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brief+History+of+Mathematics%3A+A+Promenade+Through+the+Civilizations+of+Our+World&rft.pub=Springer+Nature&rft.date=2023&rft.isbn=978-3-031-26841-0&rft.aulast=Cai&rft.aufirst=Tianxin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDFLNEAAAQBAJ%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCampbell2012" class="citation book cs1">Campbell, Stephen R. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yoEFp-Q2OXIC&pg=PT33">"Understanding Elementary Number Theory in Relation to Arithmetic and Algebra"</a>. In Zazkis, Rina; Campbell, Stephen R. (eds.). <i>Number Theory in Mathematics Education: Perspectives and Prospects</i>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-50143-2" title="Special:BookSources/978-1-136-50143-2"><bdi>978-1-136-50143-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Understanding+Elementary+Number+Theory+in+Relation+to+Arithmetic+and+Algebra&rft.btitle=Number+Theory+in+Mathematics+Education%3A+Perspectives+and+Prospects&rft.pub=Routledge&rft.date=2012&rft.isbn=978-1-136-50143-2&rft.aulast=Campbell&rft.aufirst=Stephen+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyoEFp-Q2OXIC%26pg%3DPT33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCampbell-KellyCroarkenFloodRobson2007" class="citation book cs1">Campbell-Kelly, Martin; Croarken, Mary; Flood, Raymond; Robson, Eleanor (2007). <i>The History of Mathematical Tables: From Sumer to Spreadsheets</i>. OUP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850841-0" title="Special:BookSources/978-0-19-850841-0"><bdi>978-0-19-850841-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematical+Tables%3A+From+Sumer+to+Spreadsheets&rft.pub=OUP&rft.date=2007&rft.isbn=978-0-19-850841-0&rft.aulast=Campbell-Kelly&rft.aufirst=Martin&rft.au=Croarken%2C+Mary&rft.au=Flood%2C+Raymond&rft.au=Robson%2C+Eleanor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCaprioAveniMukherjee2022" class="citation journal cs1">Caprio, Michele; Aveni, Andrea; Mukherjee, Sayan (2022). "Concerning Three Classes of Non-Diophantine Arithmetics". <i>Involve, A Journal of Mathematics</i>. <b>15</b> (5): <span class="nowrap">763–</span>774. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2102.04197">2102.04197</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Finvolve.2022.15.763">10.2140/involve.2022.15.763</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:231847291">231847291</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Involve%2C+A+Journal+of+Mathematics&rft.atitle=Concerning+Three+Classes+of+Non-Diophantine+Arithmetics&rft.volume=15&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E763-%3C%2Fspan%3E774&rft.date=2022&rft_id=info%3Aarxiv%2F2102.04197&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A231847291%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2140%2Finvolve.2022.15.763&rft.aulast=Caprio&rft.aufirst=Michele&rft.au=Aveni%2C+Andrea&rft.au=Mukherjee%2C+Sayan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCarraherSchliemann2015" class="citation book cs1">Carraher, David W.; Schliemann, Analucia D. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lpFGCgAAQBAJ&pg=PA197">"Powerful Ideas in Elementary School Mathematics"</a>. In <a href="/wiki/Lyn_English" title="Lyn English">English, Lyn D.</a>; Kirshner, David (eds.). <i>Handbook of International Research in Mathematics Education</i>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-134-62664-9" title="Special:BookSources/978-1-134-62664-9"><bdi>978-1-134-62664-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Powerful+Ideas+in+Elementary+School+Mathematics&rft.btitle=Handbook+of+International+Research+in+Mathematics+Education&rft.pub=Routledge&rft.date=2015&rft.isbn=978-1-134-62664-9&rft.aulast=Carraher&rft.aufirst=David+W.&rft.au=Schliemann%2C+Analucia+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlpFGCgAAQBAJ%26pg%3DPA197&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCavanagh2017" class="citation book cs1">Cavanagh, Joseph (2017). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=pAlEDwAAQBAJ&pg=SA5-PA32">"6. Fixed-Point Multiplication"</a></span>. <i>Computer Arithmetic and Verilog HDL Fundamentals</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-83411-7" title="Special:BookSources/978-1-351-83411-7"><bdi>978-1-351-83411-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6.+Fixed-Point+Multiplication&rft.btitle=Computer+Arithmetic+and+Verilog+HDL+Fundamentals&rft.pub=CRC+Press&rft.date=2017&rft.isbn=978-1-351-83411-7&rft.aulast=Cavanagh&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpAlEDwAAQBAJ%26pg%3DSA5-PA32&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChakravertyRout2022" class="citation book cs1">Chakraverty, Snehashish; Rout, Saudamini (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D4ByEAAAQBAJ"><i>Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-02424-5" title="Special:BookSources/978-3-031-02424-5"><bdi>978-3-031-02424-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Affine+Arithmetic+Based+Solution+of+Uncertain+Static+and+Dynamic+Problems&rft.pub=Springer+Nature&rft.date=2022&rft.isbn=978-3-031-02424-5&rft.aulast=Chakraverty&rft.aufirst=Snehashish&rft.au=Rout%2C+Saudamini&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DD4ByEAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChemlaKellerProust2023" class="citation book cs1">Chemla, Karine; Keller, Agathe; Proust, Christine (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=afKkEAAAQBAJ&pg=PA47"><i>Cultures of Computation and Quantification in the Ancient World: Numbers, Measurements, and Operations in Documents from Mesopotamia, China and South Asia</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-98361-1" title="Special:BookSources/978-3-030-98361-1"><bdi>978-3-030-98361-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cultures+of+Computation+and+Quantification+in+the+Ancient+World%3A+Numbers%2C+Measurements%2C+and+Operations+in+Documents+from+Mesopotamia%2C+China+and+South+Asia&rft.pub=Springer+Nature&rft.date=2023&rft.isbn=978-3-030-98361-1&rft.aulast=Chemla&rft.aufirst=Karine&rft.au=Keller%2C+Agathe&rft.au=Proust%2C+Christine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DafKkEAAAQBAJ%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCignoniCossu2016" class="citation book cs1">Cignoni, Gioanni A.; Cossu, Giovanni A. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LridDQAAQBAJ&pg=PA103">"The Global Virtual Museum of Information Science & Technology, a Project Idea"</a>. In Tatnall, Arthur; Leslie, Christopher (eds.). <i>International Communities of Invention and Innovation: IFIP WG 9.7 International Conference on the History of Computing, HC 2016, Brooklyn, NY, USA, May 25-29, 2016, Revised Selected Papers</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-49463-0" title="Special:BookSources/978-3-319-49463-0"><bdi>978-3-319-49463-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Global+Virtual+Museum+of+Information+Science+%26+Technology%2C+a+Project+Idea&rft.btitle=International+Communities+of+Invention+and+Innovation%3A+IFIP+WG+9.7+International+Conference+on+the+History+of+Computing%2C+HC+2016%2C+Brooklyn%2C+NY%2C+USA%2C+May+25-29%2C+2016%2C+Revised+Selected+Papers&rft.pub=Springer&rft.date=2016&rft.isbn=978-3-319-49463-0&rft.aulast=Cignoni&rft.aufirst=Gioanni+A.&rft.au=Cossu%2C+Giovanni+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLridDQAAQBAJ%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCohen2003" class="citation book cs1">Cohen, Joel S. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=URK2DwAAQBAJ&pg=PA37"><i>Computer Algebra and Symbolic Computation: Mathematical Methods</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-6370-1" title="Special:BookSources/978-1-4398-6370-1"><bdi>978-1-4398-6370-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Algebra+and+Symbolic+Computation%3A+Mathematical+Methods&rft.pub=CRC+Press&rft.date=2003&rft.isbn=978-1-4398-6370-1&rft.aulast=Cohen&rft.aufirst=Joel+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DURK2DwAAQBAJ%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFColyvan2023" class="citation web cs1">Colyvan, Mark (2023). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/mathphil-indis/">"Indispensability Arguments in the Philosophy of Mathematics"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. Metaphysics Research Lab, Stanford University<span class="reference-accessdate">. Retrieved <span class="nowrap">20 March</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Indispensability+Arguments+in+the+Philosophy+of+Mathematics&rft.date=2023&rft.aulast=Colyvan&rft.aufirst=Mark&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fmathphil-indis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConfrey1994" class="citation book cs1">Confrey, Jere (1994). "Splitting, Similarity, and Rate of Change: A New Approach to Multiplication and Exponential Functions". In Harel, Guershon; Confrey, Jere (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4MwZlzgvaGYC"><i>The Development of Multiplicative Reasoning in the Learning of Mathematics</i></a>. State University of New York Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4384-0580-3" title="Special:BookSources/978-1-4384-0580-3"><bdi>978-1-4384-0580-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Splitting%2C+Similarity%2C+and+Rate+of+Change%3A+A+New+Approach+to+Multiplication+and+Exponential+Functions&rft.btitle=The+Development+of+Multiplicative+Reasoning+in+the+Learning+of+Mathematics&rft.pub=State+University+of+New+York+Press&rft.date=1994&rft.isbn=978-1-4384-0580-3&rft.aulast=Confrey&rft.aufirst=Jere&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4MwZlzgvaGYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConradieGoranko2015" class="citation book cs1">Conradie, Willem; Goranko, Valentin (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268"><i>Logic and Discrete Mathematics: A Concise Introduction</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-76109-0" title="Special:BookSources/978-1-118-76109-0"><bdi>978-1-118-76109-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+and+Discrete+Mathematics%3A+A+Concise+Introduction&rft.pub=John+Wiley+%26+Sons&rft.date=2015&rft.isbn=978-1-118-76109-0&rft.aulast=Conradie&rft.aufirst=Willem&rft.au=Goranko%2C+Valentin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DD3jCCAAAQBAJ%26pg%3DPA268&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCryer2014" class="citation book cs1">Cryer, C. W. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_x3pAwAAQBAJ&pg=PA450"><i>A Math Primer for Engineers</i></a>. IOS Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61499-299-8" title="Special:BookSources/978-1-61499-299-8"><bdi>978-1-61499-299-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Math+Primer+for+Engineers&rft.pub=IOS+Press&rft.date=2014&rft.isbn=978-1-61499-299-8&rft.aulast=Cryer&rft.aufirst=C.+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_x3pAwAAQBAJ%26pg%3DPA450&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCunningham2016" class="citation book cs1">Cunningham, Daniel W. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fdigDAAAQBAJ&pg=PA83"><i>Set Theory: A First Course</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-316-68204-3" title="Special:BookSources/978-1-316-68204-3"><bdi>978-1-316-68204-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory%3A+A+First+Course&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=978-1-316-68204-3&rft.aulast=Cunningham&rft.aufirst=Daniel+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfdigDAAAQBAJ%26pg%3DPA83&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCurley2011" class="citation book cs1">Curley, Robert (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EdGbAAAAQBAJ&pg=PA5"><i>Computing: From the Abacus to the iPad</i></a>. Britannica Educational Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61530-707-4" title="Special:BookSources/978-1-61530-707-4"><bdi>978-1-61530-707-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computing%3A+From+the+Abacus+to+the+iPad&rft.pub=Britannica+Educational+Publishing&rft.date=2011&rft.isbn=978-1-61530-707-4&rft.aulast=Curley&rft.aufirst=Robert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEdGbAAAAQBAJ%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCuytPetersenVerdonkWaadeland2008" class="citation book cs1"><a href="/wiki/Annie_Cuyt" title="Annie Cuyt">Cuyt, Annie A. M.</a>; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182"><i>Handbook of Continued Fractions for Special Functions</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-6949-9" title="Special:BookSources/978-1-4020-6949-9"><bdi>978-1-4020-6949-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Continued+Fractions+for+Special+Functions&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008&rft.isbn=978-1-4020-6949-9&rft.aulast=Cuyt&rft.aufirst=Annie+A.+M.&rft.au=Petersen%2C+Vigdis&rft.au=Verdonk%2C+Brigitte&rft.au=Waadeland%2C+Haakon&rft.au=Jones%2C+William+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDQtpJaEs4NIC%26pg%3DPA182&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavisGouldingSuggate2017" class="citation book cs1">Davis, Andrew; Goulding, Maria; Suggate, Jennifer (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7T8lDwAAQBAJ&pg=PA11"><i>Mathematical Knowledge for Primary Teachers</i></a>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-317-21901-9" title="Special:BookSources/978-1-317-21901-9"><bdi>978-1-317-21901-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Knowledge+for+Primary+Teachers&rft.pub=Taylor+%26+Francis&rft.date=2017&rft.isbn=978-1-317-21901-9&rft.aulast=Davis&rft.aufirst=Andrew&rft.au=Goulding%2C+Maria&rft.au=Suggate%2C+Jennifer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7T8lDwAAQBAJ%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDe_CruzNethSchlimm2010" class="citation book cs1">De Cruz, Helen; Neth, Hansjörg; Schlimm, Dirk (2010). "The Cognitive Basis of Arithmetic". In Löwe, Benedikt; Müller, Thomas (eds.). <a rel="nofollow" class="external text" href="https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content"><i>PhiMSAMP: Philosophy of Mathematics : Sociological Aspsects and Mathematical Practice</i></a>. College Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-904987-95-6" title="Special:BookSources/978-1-904987-95-6"><bdi>978-1-904987-95-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Cognitive+Basis+of+Arithmetic&rft.btitle=PhiMSAMP%3A+Philosophy+of+Mathematics+%3A+Sociological+Aspsects+and+Mathematical+Practice&rft.pub=College+Publications&rft.date=2010&rft.isbn=978-1-904987-95-6&rft.aulast=De+Cruz&rft.aufirst=Helen&rft.au=Neth%2C+Hansj%C3%B6rg&rft.au=Schlimm%2C+Dirk&rft_id=https%3A%2F%2Fkops.uni-konstanz.de%2Fserver%2Fapi%2Fcore%2Fbitstreams%2F08a99b71-10aa-4c26-8cd0-9fff4e1e5427%2Fcontent&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDowker2019" class="citation book cs1">Dowker, Ann (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GQaQDwAAQBAJ&pg=PA114"><i>Individual Differences in Arithmetic: Implications for Psychology, Neuroscience and Education</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-317-62743-2" title="Special:BookSources/978-1-317-62743-2"><bdi>978-1-317-62743-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Individual+Differences+in+Arithmetic%3A+Implications+for+Psychology%2C+Neuroscience+and+Education&rft.pub=Routledge&rft.date=2019&rft.isbn=978-1-317-62743-2&rft.aulast=Dowker&rft.aufirst=Ann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGQaQDwAAQBAJ%26pg%3DPA114&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDreeben-Irimia2010" class="citation book cs1">Dreeben-Irimia, Olga (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L3gqsXsoiiAC&pg=PA102"><i>Patient Education in Rehabilitation</i></a>. Jones & Bartlett Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4496-1775-2" title="Special:BookSources/978-1-4496-1775-2"><bdi>978-1-4496-1775-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Patient+Education+in+Rehabilitation&rft.pub=Jones+%26+Bartlett+Publishers&rft.date=2010&rft.isbn=978-1-4496-1775-2&rft.aulast=Dreeben-Irimia&rft.aufirst=Olga&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DL3gqsXsoiiAC%26pg%3DPA102&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDrosg2007" class="citation book cs1">Drosg, Manfred (2007). <i>Dealing with Uncertainties</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-29606-5" title="Special:BookSources/978-3-540-29606-5"><bdi>978-3-540-29606-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dealing+with+Uncertainties&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-29606-5&rft.aulast=Drosg&rft.aufirst=Manfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDuffy2018" class="citation book cs1">Duffy, Daniel J. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BTttDwAAQBAJ&pg=PT1225"><i>Financial Instrument Pricing Using C++</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-17048-8" title="Special:BookSources/978-1-119-17048-8"><bdi>978-1-119-17048-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Instrument+Pricing+Using+C%2B%2B&rft.pub=John+Wiley+%26+Sons&rft.date=2018&rft.isbn=978-1-119-17048-8&rft.aulast=Duffy&rft.aufirst=Daniel+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBTttDwAAQBAJ%26pg%3DPT1225&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDuverney2010" class="citation book cs1">Duverney, Daniel (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sr5S9oN1xPAC&pg=PR5"><i>Number Theory: An Elementary Introduction Through Diophantine Problems</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4307-46-8" title="Special:BookSources/978-981-4307-46-8"><bdi>978-981-4307-46-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory%3A+An+Elementary+Introduction+Through+Diophantine+Problems&rft.pub=World+Scientific&rft.date=2010&rft.isbn=978-981-4307-46-8&rft.aulast=Duverney&rft.aufirst=Daniel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dsr5S9oN1xPAC%26pg%3DPR5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEbbyHulbertBroadhead2020" class="citation book cs1">Ebby, Caroline B.; Hulbert, Elizabeth T.; Broadhead, Rachel M. (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=shEHEAAAQBAJ&pg=PA24"><i>A Focus on Addition and Subtraction: Bringing Mathematics Education Research to the Classroom</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-000-22087-2" title="Special:BookSources/978-1-000-22087-2"><bdi>978-1-000-22087-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Focus+on+Addition+and+Subtraction%3A+Bringing+Mathematics+Education+Research+to+the+Classroom&rft.pub=Routledge&rft.date=2020&rft.isbn=978-1-000-22087-2&rft.aulast=Ebby&rft.aufirst=Caroline+B.&rft.au=Hulbert%2C+Elizabeth+T.&rft.au=Broadhead%2C+Rachel+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DshEHEAAAQBAJ%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEmersonBabtie2014" class="citation book cs1">Emerson, Jane; Babtie, Patricia (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NQ-aBQAAQBAJ&pg=PA147"><i>The Dyscalculia Solution: Teaching Number Sense</i></a>. Bloomsbury Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4729-2099-7" title="Special:BookSources/978-1-4729-2099-7"><bdi>978-1-4729-2099-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Dyscalculia+Solution%3A+Teaching+Number+Sense&rft.pub=Bloomsbury+Publishing&rft.date=2014&rft.isbn=978-1-4729-2099-7&rft.aulast=Emerson&rft.aufirst=Jane&rft.au=Babtie%2C+Patricia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNQ-aBQAAQBAJ%26pg%3DPA147&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFErikssonEstepJohnson2013" class="citation book cs1">Eriksson, Kenneth; Estep, Donald; Johnson, Claes (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FD8mBQAAQBAJ&pg=PA474"><i>Applied Mathematics: Body and Soul: Volume 2: Integrals and Geometry in IRn</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-05798-8" title="Special:BookSources/978-3-662-05798-8"><bdi>978-3-662-05798-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Mathematics%3A+Body+and+Soul%3A+Volume+2%3A+Integrals+and+Geometry+in+IRn&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-3-662-05798-8&rft.aulast=Eriksson&rft.aufirst=Kenneth&rft.au=Estep%2C+Donald&rft.au=Johnson%2C+Claes&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFD8mBQAAQBAJ%26pg%3DPA474&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFarmer2023" class="citation book cs1">Farmer, William M. (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VfOkEAAAQBAJ&pg=PA139"><i>Simple Type Theory: A Practical Logic for Expressing and Reasoning About Mathematical Ideas</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-21112-6" title="Special:BookSources/978-3-031-21112-6"><bdi>978-3-031-21112-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Simple+Type+Theory%3A+A+Practical+Logic+for+Expressing+and+Reasoning+About+Mathematical+Ideas&rft.pub=Springer+Nature&rft.date=2023&rft.isbn=978-3-031-21112-6&rft.aulast=Farmer&rft.aufirst=William+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVfOkEAAAQBAJ%26pg%3DPA139&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFerreiros2013" class="citation book cs1">Ferreiros, Jose (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-WL0BwAAQBAJ&pg=PA251"><i>Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics</i></a>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-5049-0" title="Special:BookSources/978-3-0348-5049-0"><bdi>978-3-0348-5049-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Labyrinth+of+Thought%3A+A+History+of+Set+Theory+and+Its+Role+in+Modern+Mathematics&rft.pub=Birkh%C3%A4user&rft.date=2013&rft.isbn=978-3-0348-5049-0&rft.aulast=Ferreiros&rft.aufirst=Jose&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-WL0BwAAQBAJ%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGallistelGelman2005" class="citation book cs1">Gallistel, C. R.; Gelman, R. (2005). "Mathematical Cognition". In Holyoak, K. J.; Morrison, R. G. (eds.). <i>The Cambridge Handbook of Thinking and Reasoning</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53101-6" title="Special:BookSources/978-0-521-53101-6"><bdi>978-0-521-53101-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematical+Cognition&rft.btitle=The+Cambridge+Handbook+of+Thinking+and+Reasoning&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-53101-6&rft.aulast=Gallistel&rft.aufirst=C.+R.&rft.au=Gelman%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGeary2006" class="citation book cs1">Geary, David C. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796">"Development of Mathematical Understanding"</a>. In Damon, William; Lerner, Richard M.; Kuhn, Deanna; Siegler, Robert S. (eds.). <i>Handbook of Child Psychology, Cognition, Perception, and Language</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-05054-5" title="Special:BookSources/978-0-470-05054-5"><bdi>978-0-470-05054-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Development+of+Mathematical+Understanding&rft.btitle=Handbook+of+Child+Psychology%2C+Cognition%2C+Perception%2C+and+Language&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-470-05054-5&rft.aulast=Geary&rft.aufirst=David+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbLZyrZHd1QkC%26pg%3DPA796&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGellertHellwichKästnerKüstner2012" class="citation book cs1">Gellert, W.; Hellwich, M.; Kästner, H.; Küstner, H. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA33"><i>The VNR Concise Encyclopedia of Mathematics</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-011-6982-0" title="Special:BookSources/978-94-011-6982-0"><bdi>978-94-011-6982-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+VNR+Concise+Encyclopedia+of+Mathematics&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-94-011-6982-0&rft.aulast=Gellert&rft.aufirst=W.&rft.au=Hellwich%2C+M.&rft.au=K%C3%A4stner%2C+H.&rft.au=K%C3%BCstner%2C+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1jH7CAAAQBAJ%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGerardiGoetteMeier2013" class="citation journal cs1">Gerardi, Kristopher; Goette, Lorenz; Meier, Stephan (2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3710828">"Numerical Ability Predicts Mortgage Default"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>110</b> (28): <span class="nowrap">11267–</span>11271. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PNAS..11011267G">2013PNAS..11011267G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1220568110">10.1073/pnas.1220568110</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3710828">3710828</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23798401">23798401</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Numerical+Ability+Predicts+Mortgage+Default&rft.volume=110&rft.issue=28&rft.pages=%3Cspan+class%3D%22nowrap%22%3E11267-%3C%2Fspan%3E11271&rft.date=2013&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3710828%23id-name%3DPMC&rft_id=info%3Apmid%2F23798401&rft_id=info%3Adoi%2F10.1073%2Fpnas.1220568110&rft_id=info%3Abibcode%2F2013PNAS..11011267G&rft.aulast=Gerardi&rft.aufirst=Kristopher&rft.au=Goette%2C+Lorenz&rft.au=Meier%2C+Stephan&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3710828&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoodstein2014" class="citation book cs1">Goodstein, R. L. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EHjiBQAAQBAJ&pg=PA33"><i>Fundamental Concepts of Mathematics</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4831-5405-3" title="Special:BookSources/978-1-4831-5405-3"><bdi>978-1-4831-5405-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Concepts+of+Mathematics&rft.pub=Elsevier&rft.date=2014&rft.isbn=978-1-4831-5405-3&rft.aulast=Goodstein&rft.aufirst=R.+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEHjiBQAAQBAJ%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriceKempMortonGrace2023" class="citation journal cs1">Grice, Matt; Kemp, Simon; Morton, Nicola J.; Grace, Randolph C. (2023). <a rel="nofollow" class="external text" href="https://psyarxiv.com/myq4n/">"The Psychological Scaffolding of Arithmetic"</a>. <i>Psychological Review</i>. <b>131</b> (2): <span class="nowrap">494–</span>522. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Frev0000431">10.1037/rev0000431</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37358523">37358523</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:259251163">259251163</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Review&rft.atitle=The+Psychological+Scaffolding+of+Arithmetic&rft.volume=131&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E494-%3C%2Fspan%3E522&rft.date=2023&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A259251163%23id-name%3DS2CID&rft_id=info%3Apmid%2F37358523&rft_id=info%3Adoi%2F10.1037%2Frev0000431&rft.aulast=Grice&rft.aufirst=Matt&rft.au=Kemp%2C+Simon&rft.au=Morton%2C+Nicola+J.&rft.au=Grace%2C+Randolph+C.&rft_id=https%3A%2F%2Fpsyarxiv.com%2Fmyq4n%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrigorieva2018" class="citation book cs1">Grigorieva, Ellina (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mEpjDwAAQBAJ&pg=PR8"><i>Methods of Solving Number Theory Problems</i></a>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-90915-8" title="Special:BookSources/978-3-319-90915-8"><bdi>978-3-319-90915-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Solving+Number+Theory+Problems&rft.pub=Birkh%C3%A4user&rft.date=2018&rft.isbn=978-3-319-90915-8&rft.aulast=Grigorieva&rft.aufirst=Ellina&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmEpjDwAAQBAJ%26pg%3DPR8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriffin1935" class="citation journal cs1">Griffin, Carroll W. (1935). "Significant Figures". <i>National Mathematics Magazine</i>. <b>10</b> (1): <span class="nowrap">20–</span>24. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3028249">10.2307/3028249</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3028249">3028249</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=National+Mathematics+Magazine&rft.atitle=Significant+Figures&rft.volume=10&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E20-%3C%2Fspan%3E24&rft.date=1935&rft_id=info%3Adoi%2F10.2307%2F3028249&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3028249%23id-name%3DJSTOR&rft.aulast=Griffin&rft.aufirst=Carroll+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGupta2019" class="citation book cs1">Gupta, Rajesh Kumar (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vcmcDwAAQBAJ&pg=PA3"><i>Numerical Methods: Fundamentals and Applications</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-68660-0" title="Special:BookSources/978-1-108-68660-0"><bdi>978-1-108-68660-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods%3A+Fundamentals+and+Applications&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=978-1-108-68660-0&rft.aulast=Gupta&rft.aufirst=Rajesh+Kumar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvcmcDwAAQBAJ%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHafstrom2013" class="citation book cs1">Hafstrom, John Edward (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA6"><i>Basic Concepts in Modern Mathematics</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-31627-7" title="Special:BookSources/978-0-486-31627-7"><bdi>978-0-486-31627-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Concepts+in+Modern+Mathematics&rft.pub=Courier+Corporation&rft.date=2013&rft.isbn=978-0-486-31627-7&rft.aulast=Hafstrom&rft.aufirst=John+Edward&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dmj_DAgAAQBAJ%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHamiltonLandin2018" class="citation book cs1">Hamilton, Norman T.; Landin, Joseph (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA133"><i>Set Theory: The Structure of Arithmetic</i></a>. Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-83047-6" title="Special:BookSources/978-0-486-83047-6"><bdi>978-0-486-83047-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory%3A+The+Structure+of+Arithmetic&rft.pub=Courier+Dover+Publications&rft.date=2018&rft.isbn=978-0-486-83047-6&rft.aulast=Hamilton&rft.aufirst=Norman+T.&rft.au=Landin%2C+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9aReDwAAQBAJ%26pg%3DPA133&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHart2011" class="citation book cs1">Hart, Roger (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zLPm3xE2qWgC&pg=PA69"><i>The Chinese Roots of Linear Algebra</i></a>. JHU Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-9958-4" title="Special:BookSources/978-0-8018-9958-4"><bdi>978-0-8018-9958-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Chinese+Roots+of+Linear+Algebra&rft.pub=JHU+Press&rft.date=2011&rft.isbn=978-0-8018-9958-4&rft.aulast=Hart&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzLPm3xE2qWgC%26pg%3DPA69&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHaylockCockburn2008" class="citation book cs1">Haylock, Derek; Cockburn, Anne D. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hgAr3maZeQUC&pg=PA49"><i>Understanding Mathematics for Young Children: A Guide for Foundation Stage and Lower Primary Teachers</i></a>. SAGE. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4462-0497-9" title="Special:BookSources/978-1-4462-0497-9"><bdi>978-1-4462-0497-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Mathematics+for+Young+Children%3A+A+Guide+for+Foundation+Stage+and+Lower+Primary+Teachers&rft.pub=SAGE&rft.date=2008&rft.isbn=978-1-4462-0497-9&rft.aulast=Haylock&rft.aufirst=Derek&rft.au=Cockburn%2C+Anne+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhgAr3maZeQUC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHC_staff2022" class="citation web cs1">HC staff (2022). <a rel="nofollow" class="external text" href="https://www.ahdictionary.com/word/search.html?q=numeral&submit.x=58&submit.y=14">"Numeral"</a>. <i>American Heritage Dictionary</i>. HarperCollins<span class="reference-accessdate">. Retrieved <span class="nowrap">11 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=American+Heritage+Dictionary&rft.atitle=Numeral&rft.date=2022&rft.au=HC+staff&rft_id=https%3A%2F%2Fwww.ahdictionary.com%2Fword%2Fsearch.html%3Fq%3Dnumeral%26submit.x%3D58%26submit.y%3D14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHC_staff2022a" class="citation web cs1">HC staff (2022a). <a rel="nofollow" class="external text" href="https://www.ahdictionary.com/word/search.html?q=number+system&submit.x=58&submit.y=14">"Number System"</a>. <i>American Heritage Dictionary</i>. HarperCollins<span class="reference-accessdate">. Retrieved <span class="nowrap">11 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=American+Heritage+Dictionary&rft.atitle=Number+System&rft.date=2022&rft.au=HC+staff&rft_id=https%3A%2F%2Fwww.ahdictionary.com%2Fword%2Fsearch.html%3Fq%3Dnumber%2Bsystem%26submit.x%3D58%26submit.y%3D14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHC_staff2022b" class="citation web cs1">HC staff (2022b). <a rel="nofollow" class="external text" href="https://www.ahdictionary.com/word/search.html?q=arithmetic&submit.x=58&submit.y=14">"Arithmetic"</a>. <i>American Heritage Dictionary</i>. HarperCollins<span class="reference-accessdate">. Retrieved <span class="nowrap">19 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=American+Heritage+Dictionary&rft.atitle=Arithmetic&rft.date=2022&rft.au=HC+staff&rft_id=https%3A%2F%2Fwww.ahdictionary.com%2Fword%2Fsearch.html%3Fq%3Darithmetic%26submit.x%3D58%26submit.y%3D14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHigham2002" class="citation book cs1">Higham, Nicholas (2002). <a rel="nofollow" class="external text" href="http://eprints.maths.manchester.ac.uk/238/4/asna2_cover.pdf"><i>Accuracy and Stability of Numerical Algorithms</i></a> <span class="cs1-format">(PDF)</span> (2nd ed.). SIAM. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9780898718027">10.1137/1.9780898718027</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-521-7" title="Special:BookSources/978-0-89871-521-7"><bdi>978-0-89871-521-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Accuracy+and+Stability+of+Numerical+Algorithms&rft.edition=2nd&rft.pub=SIAM&rft.date=2002&rft_id=info%3Adoi%2F10.1137%2F1.9780898718027&rft.isbn=978-0-89871-521-7&rft.aulast=Higham&rft.aufirst=Nicholas&rft_id=http%3A%2F%2Feprints.maths.manchester.ac.uk%2F238%2F4%2Fasna2_cover.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHindry2011" class="citation book cs1">Hindry, Marc (2011). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-1-4471-2131-2"><i>Arithmetics</i></a>. Universitext. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4471-2131-2">10.1007/978-1-4471-2131-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-2130-5" title="Special:BookSources/978-1-4471-2130-5"><bdi>978-1-4471-2130-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetics&rft.series=Universitext&rft.pub=Springer&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2F978-1-4471-2131-2&rft.isbn=978-1-4471-2130-5&rft.aulast=Hindry&rft.aufirst=Marc&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-1-4471-2131-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHodgkin2013" class="citation book cs1">Hodgkin, Luke (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nSO5iMujRUYC&pg=PR168"><i>A History of Mathematics: From Mesopotamia to Modernity</i></a>. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-166436-6" title="Special:BookSources/978-0-19-166436-6"><bdi>978-0-19-166436-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3A+From+Mesopotamia+to+Modernity&rft.pub=OUP+Oxford&rft.date=2013&rft.isbn=978-0-19-166436-6&rft.aulast=Hodgkin&rft.aufirst=Luke&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnSO5iMujRUYC%26pg%3DPR168&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHoffmanFrankel2018" class="citation book cs1">Hoffman, Joe D.; Frankel, Steven (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=F5K3DwAAQBAJ&pg=PA161"><i>Numerical Methods for Engineers and Scientists</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-7060-0" title="Special:BookSources/978-1-4822-7060-0"><bdi>978-1-4822-7060-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+for+Engineers+and+Scientists&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-4822-7060-0&rft.aulast=Hoffman&rft.aufirst=Joe+D.&rft.au=Frankel%2C+Steven&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DF5K3DwAAQBAJ%26pg%3DPA161&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHofweber2016" class="citation book cs1">Hofweber, Thomas (2016). "The Philosophy of Arithmetic". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EYDADAAAQBAJ"><i>Ontology and the Ambitions of Metaphysics</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-876983-5" title="Special:BookSources/978-0-19-876983-5"><bdi>978-0-19-876983-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Philosophy+of+Arithmetic&rft.btitle=Ontology+and+the+Ambitions+of+Metaphysics&rft.pub=Oxford+University+Press&rft.date=2016&rft.isbn=978-0-19-876983-5&rft.aulast=Hofweber&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEYDADAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHorsten2023" class="citation web cs1">Horsten, Leon (2023). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/philosophy-mathematics/">"Philosophy of Mathematics"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. Metaphysics Research Lab, Stanford University<span class="reference-accessdate">. Retrieved <span class="nowrap">22 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Philosophy+of+Mathematics&rft.date=2023&rft.aulast=Horsten&rft.aufirst=Leon&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fphilosophy-mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHuaFeng2020" class="citation book cs1">Hua, Jueming; Feng, Lisheng (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6_sOEAAAQBAJ&pg=PA119"><i>Thirty Great Inventions of China: From Millet Agriculture to Artemisinin</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-15-6525-0" title="Special:BookSources/978-981-15-6525-0"><bdi>978-981-15-6525-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thirty+Great+Inventions+of+China%3A+From+Millet+Agriculture+to+Artemisinin&rft.pub=Springer+Nature&rft.date=2020&rft.isbn=978-981-15-6525-0&rft.aulast=Hua&rft.aufirst=Jueming&rft.au=Feng%2C+Lisheng&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6_sOEAAAQBAJ%26pg%3DPA119&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHusserlWillard2012" class="citation book cs1">Husserl, Edmund; Willard, Dallas (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lxftCAAAQBAJ&pg=PR44">"Translator's Introduction"</a>. <i>Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887–1901</i>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-010-0060-4" title="Special:BookSources/978-94-010-0060-4"><bdi>978-94-010-0060-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Translator%27s+Introduction&rft.btitle=Philosophy+of+Arithmetic%3A+Psychological+and+Logical+Investigations+with+Supplementary+Texts+from+1887%E2%80%931901&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-94-010-0060-4&rft.aulast=Husserl&rft.aufirst=Edmund&rft.au=Willard%2C+Dallas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlxftCAAAQBAJ%26pg%3DPR44&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIgarashiAltmanFunadaKamiyama2014" class="citation book cs1">Igarashi, Yoshihide; Altman, Tom; Funada, Mariko; Kamiyama, Barbara (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA18"><i>Computing: A Historical and Technical Perspective</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-2741-3" title="Special:BookSources/978-1-4822-2741-3"><bdi>978-1-4822-2741-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computing%3A+A+Historical+and+Technical+Perspective&rft.pub=CRC+Press&rft.date=2014&rft.isbn=978-1-4822-2741-3&rft.aulast=Igarashi&rft.aufirst=Yoshihide&rft.au=Altman%2C+Tom&rft.au=Funada%2C+Mariko&rft.au=Kamiyama%2C+Barbara&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D58ySAwAAQBAJ%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFInternational_Organization_for_Standardization2019" class="citation book cs1">International Organization for Standardization (2019). <a rel="nofollow" class="external text" href="https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf"><i>ISO 80000-2: 2019 Quantities and units Part 2: Mathematics</i></a> <span class="cs1-format">(PDF)</span>. International Organization for Standardization.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=ISO+80000-2%3A+2019+Quantities+and+units+Part+2%3A+Mathematics&rft.pub=International+Organization+for+Standardization&rft.date=2019&rft.au=International+Organization+for+Standardization&rft_id=https%3A%2F%2Fcdn.standards.iteh.ai%2Fsamples%2F64973%2F329519100abd447ea0d49747258d1094%2FISO-80000-2-2019.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFITL_Education_Solutions_Limited2011" class="citation book cs1">ITL Education Solutions Limited (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28"><i>Introduction to Computer Science</i></a>. Pearson Education India. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-6030-7" title="Special:BookSources/978-81-317-6030-7"><bdi>978-81-317-6030-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Computer+Science&rft.pub=Pearson+Education+India&rft.date=2011&rft.isbn=978-81-317-6030-7&rft.au=ITL+Education+Solutions+Limited&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCsNiKdmufvYC%26pg%3DPA28&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJackson2008" class="citation book cs1">Jackson, Janna M. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tLuPAgAAQBAJ&pg=PA152">"Reading/Writing Connection"</a>. In Flippo, Rona F. (ed.). <i>Handbook of College Reading and Study Strategy Research</i>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-135-70373-8" title="Special:BookSources/978-1-135-70373-8"><bdi>978-1-135-70373-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reading%2FWriting+Connection&rft.btitle=Handbook+of+College+Reading+and+Study+Strategy+Research&rft.pub=Routledge&rft.date=2008&rft.isbn=978-1-135-70373-8&rft.aulast=Jackson&rft.aufirst=Janna+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtLuPAgAAQBAJ%26pg%3DPA152&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJena2021" class="citation book cs1">Jena, Sisir Kumar (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17"><i>C Programming: Learn to Code</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-000-46056-8" title="Special:BookSources/978-1-000-46056-8"><bdi>978-1-000-46056-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C+Programming%3A+Learn+to+Code&rft.pub=CRC+Press&rft.date=2021&rft.isbn=978-1-000-46056-8&rft.aulast=Jena&rft.aufirst=Sisir+Kumar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqRpSEAAAQBAJ%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaiserGranade2021" class="citation book cs1">Kaiser, Sarah C.; Granade, Christopher (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IxIxEAAAQBAJ&pg=PA283"><i>Learn Quantum Computing with Python and Q#: A Hands-on Approach</i></a>. Simon and Schuster. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61729-613-0" title="Special:BookSources/978-1-61729-613-0"><bdi>978-1-61729-613-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learn+Quantum+Computing+with+Python+and+Q%23%3A+A+Hands-on+Approach&rft.pub=Simon+and+Schuster&rft.date=2021&rft.isbn=978-1-61729-613-0&rft.aulast=Kaiser&rft.aufirst=Sarah+C.&rft.au=Granade%2C+Christopher&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIxIxEAAAQBAJ%26pg%3DPA283&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaratsuba2014" class="citation web cs1">Karatsuba, A. A. (2014). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Analytic_number_theory">"Analytic Number Theory"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Analytic+Number+Theory&rft.date=2014&rft.aulast=Karatsuba&rft.aufirst=A.+A.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FAnalytic_number_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaratsuba2020" class="citation web cs1">Karatsuba, A. A. (2020). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Number_theory">"Number Theory"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Number+Theory&rft.date=2020&rft.aulast=Karatsuba&rft.aufirst=A.+A.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FNumber_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKay2021" class="citation book cs1">Kay, Anthony (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA57"><i>Number Systems: A Path into Rigorous Mathematics</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-60776-9" title="Special:BookSources/978-0-429-60776-9"><bdi>978-0-429-60776-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Systems%3A+A+Path+into+Rigorous+Mathematics&rft.pub=CRC+Press&rft.date=2021&rft.isbn=978-0-429-60776-9&rft.aulast=Kay&rft.aufirst=Anthony&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Daw81EAAAQBAJ%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhanGraham2018" class="citation book cs1">Khan, Khalid; Graham, Tony Lee (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vy73DwAAQBAJ&pg=PA9"><i>Engineering Mathematics with Applications to Fire Engineering</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-59761-6" title="Special:BookSources/978-1-351-59761-6"><bdi>978-1-351-59761-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mathematics+with+Applications+to+Fire+Engineering&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-351-59761-6&rft.aulast=Khan&rft.aufirst=Khalid&rft.au=Graham%2C+Tony+Lee&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvy73DwAAQBAJ%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhattar2010" class="citation book cs1">Khattar, Dinesh (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I3rCgXwvffsC&pg=PA1"><i>The Pearson Guide To Objective Arithmetic For Competitive Examinations, 3/E</i></a>. Pearson Education India. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-2673-0" title="Special:BookSources/978-81-317-2673-0"><bdi>978-81-317-2673-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Pearson+Guide+To+Objective+Arithmetic+For+Competitive+Examinations%2C+3%2FE&rft.pub=Pearson+Education+India&rft.date=2010&rft.isbn=978-81-317-2673-0&rft.aulast=Khattar&rft.aufirst=Dinesh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DI3rCgXwvffsC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhouryLamothe2016" class="citation book cs1">Khoury, Joseph; Lamothe, Gilles (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T162DAAAQBAJ&pg=PA2"><i>Mathematics That Power Our World, The: How Is It Made?</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4730-86-0" title="Special:BookSources/978-981-4730-86-0"><bdi>978-981-4730-86-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+That+Power+Our+World%2C+The%3A+How+Is+It+Made%3F&rft.pub=World+Scientific&rft.date=2016&rft.isbn=978-981-4730-86-0&rft.aulast=Khoury&rft.aufirst=Joseph&rft.au=Lamothe%2C+Gilles&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT162DAAAQBAJ%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlaf2011" class="citation book cs1">Klaf, A. Albert (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YRNeBAAAQBAJ&pg=PA187"><i>Trigonometry Refresher</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-15104-5" title="Special:BookSources/978-0-486-15104-5"><bdi>978-0-486-15104-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry+Refresher&rft.pub=Courier+Corporation&rft.date=2011&rft.isbn=978-0-486-15104-5&rft.aulast=Klaf&rft.aufirst=A.+Albert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYRNeBAAAQBAJ%26pg%3DPA187&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKleinMoellerDresselDomahs2010" class="citation journal cs1">Klein, Elise; Moeller, Korbinian; Dressel, Katharina; Domahs, Frank; Wood, Guilherme; Willmes, Klaus; Nuerk, Hans-Christoph (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.actpsy.2010.06.002">"To Carry or Not to Carry — Is This the Question? Disentangling the Carry Effect in Multi-digit Addition"</a>. <i>Acta Psychologica</i>. <b>135</b> (1): <span class="nowrap">67–</span>76. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.actpsy.2010.06.002">10.1016/j.actpsy.2010.06.002</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20580340">20580340</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Psychologica&rft.atitle=To+Carry+or+Not+to+Carry+%E2%80%94+Is+This+the+Question%3F+Disentangling+the+Carry+Effect+in+Multi-digit+Addition&rft.volume=135&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E67-%3C%2Fspan%3E76&rft.date=2010&rft_id=info%3Adoi%2F10.1016%2Fj.actpsy.2010.06.002&rft_id=info%3Apmid%2F20580340&rft.aulast=Klein&rft.aufirst=Elise&rft.au=Moeller%2C+Korbinian&rft.au=Dressel%2C+Katharina&rft.au=Domahs%2C+Frank&rft.au=Wood%2C+Guilherme&rft.au=Willmes%2C+Klaus&rft.au=Nuerk%2C+Hans-Christoph&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.actpsy.2010.06.002&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlein2013" class="citation book cs1">Klein, Andreas (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GYpEAAAAQBAJ&pg=PA249"><i>Stream Ciphers</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-5079-4" title="Special:BookSources/978-1-4471-5079-4"><bdi>978-1-4471-5079-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stream+Ciphers&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4471-5079-4&rft.aulast=Klein&rft.aufirst=Andreas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGYpEAAAAQBAJ%26pg%3DPA249&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlein2013a" class="citation book cs1">Klein, Jacob (2013a). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EkPDAgAAQBAJ&pg=PT12"><i>Greek Mathematical Thought and the Origin of Algebra</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-31981-0" title="Special:BookSources/978-0-486-31981-0"><bdi>978-0-486-31981-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Greek+Mathematical+Thought+and+the+Origin+of+Algebra&rft.pub=Courier+Corporation&rft.date=2013&rft.isbn=978-0-486-31981-0&rft.aulast=Klein&rft.aufirst=Jacob&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEkPDAgAAQBAJ%26pg%3DPT12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKleiner2012" class="citation book cs1">Kleiner, Israel (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7NcEIteAfyYC&pg=PA255"><i>Excursions in the History of Mathematics</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-8268-2" title="Special:BookSources/978-0-8176-8268-2"><bdi>978-0-8176-8268-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Excursions+in+the+History+of+Mathematics&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-0-8176-8268-2&rft.aulast=Kleiner&rft.aufirst=Israel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7NcEIteAfyYC%26pg%3DPA255&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlose2014" class="citation book cs1">Klose, Orval M. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA110"><i>The Number Systems and Operations of Arithmetic: An Explanation of the Fundamental Principles of Mathematics Which Underlie the Understanding and Use of Arithmetic, Designed for In-Service Training of Elementary School Teachers Candidates Service Training of Elementary School Teacher Candidates</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4831-3709-4" title="Special:BookSources/978-1-4831-3709-4"><bdi>978-1-4831-3709-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Number+Systems+and+Operations+of+Arithmetic%3A+An+Explanation+of+the+Fundamental+Principles+of+Mathematics+Which+Underlie+the+Understanding+and+Use+of+Arithmetic%2C+Designed+for+In-Service+Training+of+Elementary+School+Teachers+Candidates+Service+Training+of+Elementary+School+Teacher+Candidates&rft.pub=Elsevier&rft.date=2014&rft.isbn=978-1-4831-3709-4&rft.aulast=Klose&rft.aufirst=Orval+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrG7iBQAAQBAJ%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKnoblochKomatsuLiu2013" class="citation book cs1">Knobloch, Eberhard; Komatsu, Hikosaburo; Liu, Dun (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JgDFBAAAQBAJ&pg=PA123"><i>Seki, Founder of Modern Mathematics in Japan: A Commemoration on His Tercentenary</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-4-431-54273-5" title="Special:BookSources/978-4-431-54273-5"><bdi>978-4-431-54273-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Seki%2C+Founder+of+Modern+Mathematics+in+Japan%3A+A+Commemoration+on+His+Tercentenary&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-4-431-54273-5&rft.aulast=Knobloch&rft.aufirst=Eberhard&rft.au=Komatsu%2C+Hikosaburo&rft.au=Liu%2C+Dun&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJgDFBAAAQBAJ%26pg%3DPA123&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoepf2021" class="citation book cs1">Koepf, Wolfram (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rOU3EAAAQBAJ&pg=PA49"><i>Computer Algebra: An Algorithm-Oriented Introduction</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-78017-3" title="Special:BookSources/978-3-030-78017-3"><bdi>978-3-030-78017-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Algebra%3A+An+Algorithm-Oriented+Introduction&rft.pub=Springer+Nature&rft.date=2021&rft.isbn=978-3-030-78017-3&rft.aulast=Koepf&rft.aufirst=Wolfram&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrOU3EAAAQBAJ%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoetsier2018" class="citation book cs1">Koetsier, Teun (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9jF7DwAAQBAJ&pg=PA255"><i>The Ascent of GIM, the Global Intelligent Machine: A History of Production and Information Machines</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-96547-5" title="Special:BookSources/978-3-319-96547-5"><bdi>978-3-319-96547-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ascent+of+GIM%2C+the+Global+Intelligent+Machine%3A+A+History+of+Production+and+Information+Machines&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-319-96547-5&rft.aulast=Koetsier&rft.aufirst=Teun&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9jF7DwAAQBAJ%26pg%3DPA255&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoren2018" class="citation book cs1">Koren, Israel (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wUBZDwAAQBAJ&pg=PA71"><i>Computer Arithmetic Algorithms</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-6371-8" title="Special:BookSources/978-1-4398-6371-8"><bdi>978-1-4398-6371-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Arithmetic+Algorithms&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-4398-6371-8&rft.aulast=Koren&rft.aufirst=Israel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwUBZDwAAQBAJ%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKörner2019" class="citation book cs1">Körner, T. W. (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z-y2DwAAQBAJ&pg=PA109"><i>Where Do Numbers Come From?</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-77594-6" title="Special:BookSources/978-1-108-77594-6"><bdi>978-1-108-77594-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Where+Do+Numbers+Come+From%3F&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=978-1-108-77594-6&rft.aulast=K%C3%B6rner&rft.aufirst=T.+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dz-y2DwAAQBAJ%26pg%3DPA109&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKrennLorünser2023" class="citation book cs1">Krenn, Stephan; Lorünser, Thomas (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RRi2EAAAQBAJ&pg=PA8"><i>An Introduction to Secret Sharing: A Systematic Overview and Guide for Protocol Selection</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-28161-7" title="Special:BookSources/978-3-031-28161-7"><bdi>978-3-031-28161-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Secret+Sharing%3A+A+Systematic+Overview+and+Guide+for+Protocol+Selection&rft.pub=Springer+Nature&rft.date=2023&rft.isbn=978-3-031-28161-7&rft.aulast=Krenn&rft.aufirst=Stephan&rft.au=Lor%C3%BCnser%2C+Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRRi2EAAAQBAJ%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKubilyus2018" class="citation web cs1">Kubilyus, I. P. (2018). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Number_theory,_probabilistic_methods_in">"Number Theory, Probabilistic Methods in"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Number+Theory%2C+Probabilistic+Methods+in&rft.date=2018&rft.aulast=Kubilyus&rft.aufirst=I.+P.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FNumber_theory%2C_probabilistic_methods_in&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKudryavtsev2020" class="citation web cs1">Kudryavtsev, L. D. (2020). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Real_number">"Real Number"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Real+Number&rft.date=2020&rft.aulast=Kudryavtsev&rft.aufirst=L.+D.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FReal_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKupferman2015" class="citation book cs1">Kupferman, Raz (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cd87DQAAQBAJ&pg=PA92"><i>Elementary School Mathematics For Parents And Teachers - Volume 1</i></a>. World Scientific Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4699-93-8" title="Special:BookSources/978-981-4699-93-8"><bdi>978-981-4699-93-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+School+Mathematics+For+Parents+And+Teachers+-+Volume+1&rft.pub=World+Scientific+Publishing+Company&rft.date=2015&rft.isbn=978-981-4699-93-8&rft.aulast=Kupferman&rft.aufirst=Raz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCd87DQAAQBAJ%26pg%3DPA92&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKřížekSomerŠolcová2021" class="citation book cs1">Křížek, Michal; Somer, Lawrence; Šolcová, Alena (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23"><i>From Great Discoveries in Number Theory to Applications</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-83899-7" title="Special:BookSources/978-3-030-83899-7"><bdi>978-3-030-83899-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Great+Discoveries+in+Number+Theory+to+Applications&rft.pub=Springer+Nature&rft.date=2021&rft.isbn=978-3-030-83899-7&rft.aulast=K%C5%99%C3%AD%C5%BEek&rft.aufirst=Michal&rft.au=Somer%2C+Lawrence&rft.au=%C5%A0olcov%C3%A1%2C+Alena&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtklEEAAAQBAJ%26pg%3DPA23&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang2002" class="citation book cs1">Lang, Serge (2002). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-1-4613-0077-9_14">"Taylor's Formula"</a>. <i>Short Calculus: The Original Edition of "A First Course in Calculus"</i>. Undergraduate Texts in Mathematics. Springer. pp. <span class="nowrap">195–</span>210. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4613-0077-9_14">10.1007/978-1-4613-0077-9_14</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-0077-9" title="Special:BookSources/978-1-4613-0077-9"><bdi>978-1-4613-0077-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Taylor%E2%80%99s+Formula&rft.btitle=Short+Calculus%3A+The+Original+Edition+of+%22A+First+Course+in+Calculus%22&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E195-%3C%2Fspan%3E210&rft.pub=Springer&rft.date=2002&rft_id=info%3Adoi%2F10.1007%2F978-1-4613-0077-9_14&rft.isbn=978-1-4613-0077-9&rft.aulast=Lang&rft.aufirst=Serge&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-1-4613-0077-9_14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang2015" class="citation book cs1">Lang, Philippa (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wbiKDwAAQBAJ&pg=PA160"><i>Science: Antiquity and its Legacy</i></a>. Bloomsbury Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85773-955-1" title="Special:BookSources/978-0-85773-955-1"><bdi>978-0-85773-955-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science%3A+Antiquity+and+its+Legacy&rft.pub=Bloomsbury+Publishing&rft.date=2015&rft.isbn=978-0-85773-955-1&rft.aulast=Lang&rft.aufirst=Philippa&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwbiKDwAAQBAJ%26pg%3DPA160&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLange2010" class="citation book cs1">Lange, Kenneth (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AtiDhx2bsiMC&pg=PA248"><i>Numerical Analysis for Statisticians</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-5944-7" title="Special:BookSources/978-1-4419-5944-7"><bdi>978-1-4419-5944-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Analysis+for+Statisticians&rft.pub=Springer+Science+%26+Business+Media&rft.date=2010&rft.isbn=978-1-4419-5944-7&rft.aulast=Lange&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAtiDhx2bsiMC%26pg%3DPA248&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLaskiJor’danDaoustMurray2015" class="citation journal cs1">Laski, Elida V.; Jor’dan, Jamilah R.; Daoust, Carolyn; Murray, Angela K. (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F2158244015589588">"What Makes Mathematics Manipulatives Effective? Lessons From Cognitive Science and Montessori Education"</a>. <i>SAGE Open</i>. <b>5</b> (2). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F2158244015589588">10.1177/2158244015589588</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1808%2F20642">1808/20642</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11722953">11722953</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SAGE+Open&rft.atitle=What+Makes+Mathematics+Manipulatives+Effective%3F+Lessons+From+Cognitive+Science+and+Montessori+Education&rft.volume=5&rft.issue=2&rft.date=2015&rft_id=info%3Ahdl%2F1808%2F20642&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11722953%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1177%2F2158244015589588&rft.aulast=Laski&rft.aufirst=Elida+V.&rft.au=Jor%E2%80%99dan%2C+Jamilah+R.&rft.au=Daoust%2C+Carolyn&rft.au=Murray%2C+Angela+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.1177%252F2158244015589588&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLernerLerner2008" class="citation book cs1">Lerner, Brenda Wilmoth; Lerner, K. Lee, eds. (2008). "Modular Arithmetic". <a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science-and-technology/computers-and-electrical-engineering/computers-and-computing/modular-arithmetic"><i>The Gale Encyclopedia of Science</i></a> (4th ed.). Thompson Gale. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4144-2877-2" title="Special:BookSources/978-1-4144-2877-2"><bdi>978-1-4144-2877-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Modular+Arithmetic&rft.btitle=The+Gale+Encyclopedia+of+Science&rft.edition=4th&rft.pub=Thompson+Gale&rft.date=2008&rft.isbn=978-1-4144-2877-2&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience-and-technology%2Fcomputers-and-electrical-engineering%2Fcomputers-and-computing%2Fmodular-arithmetic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLiSchoenfeld2019" class="citation journal cs1">Li, Yeping; Schoenfeld, Alan H. (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1186%2Fs40594-019-0197-9">"Problematizing Teaching and Learning Mathematics as 'Given' in STEM Education"</a>. <i>International Journal of STEM Education</i>. <b>6</b> (1). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1186%2Fs40594-019-0197-9">10.1186/s40594-019-0197-9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+STEM+Education&rft.atitle=Problematizing+Teaching+and+Learning+Mathematics+as+%27Given%27+in+STEM+Education&rft.volume=6&rft.issue=1&rft.date=2019&rft_id=info%3Adoi%2F10.1186%2Fs40594-019-0197-9&rft.aulast=Li&rft.aufirst=Yeping&rft.au=Schoenfeld%2C+Alan+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1186%252Fs40594-019-0197-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLiebler2018" class="citation book cs1">Liebler, Robert A. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ozb3DwAAQBAJ&pg=PA36"><i>Basic Matrix Algebra with Algorithms and Applications</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-85287-9" title="Special:BookSources/978-0-429-85287-9"><bdi>978-0-429-85287-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Matrix+Algebra+with+Algorithms+and+Applications&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-0-429-85287-9&rft.aulast=Liebler&rft.aufirst=Robert+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOzb3DwAAQBAJ%26pg%3DPA36&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLockhart2017" class="citation book cs1">Lockhart, Paul (2017). <i>Arithmetic</i>. The Belknap Press of Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-97223-0" title="Special:BookSources/978-0-674-97223-0"><bdi>978-0-674-97223-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetic&rft.pub=The+Belknap+Press+of+Harvard+University+Press&rft.date=2017&rft.isbn=978-0-674-97223-0&rft.aulast=Lockhart&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLozano-Robledo2019" class="citation book cs1">Lozano-Robledo, Álvaro (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13"><i>Number Theory and Geometry: An Introduction to Arithmetic Geometry</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-5016-8" title="Special:BookSources/978-1-4704-5016-8"><bdi>978-1-4704-5016-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+and+Geometry%3A+An+Introduction+to+Arithmetic+Geometry&rft.pub=American+Mathematical+Soc.&rft.date=2019&rft.isbn=978-1-4704-5016-8&rft.aulast=Lozano-Robledo&rft.aufirst=%C3%81lvaro&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DESiODwAAQBAJ%26pg%3DPR13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLudererNollauVetters2013" class="citation book cs1">Luderer, Bernd; Nollau, Volker; Vetters, Klaus (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rSf0CAAAQBAJ&pg=PA9"><i>Mathematical Formulas for Economists</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-12431-4" title="Special:BookSources/978-3-662-12431-4"><bdi>978-3-662-12431-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Formulas+for+Economists&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-3-662-12431-4&rft.aulast=Luderer&rft.aufirst=Bernd&rft.au=Nollau%2C+Volker&rft.au=Vetters%2C+Klaus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrSf0CAAAQBAJ%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLützen2023" class="citation book cs1">Lützen, Jesper (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=joikEAAAQBAJ&pg=PA19"><i>A History of Mathematical Impossibility</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-286739-1" title="Special:BookSources/978-0-19-286739-1"><bdi>978-0-19-286739-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Impossibility&rft.pub=Oxford+University+Press&rft.date=2023&rft.isbn=978-0-19-286739-1&rft.aulast=L%C3%BCtzen&rft.aufirst=Jesper&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjoikEAAAQBAJ%26pg%3DPA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLustick1997" class="citation journal cs1">Lustick, David (1997). "The Numbers Game: Playing up the importance of significant figures and scientific notation". <i>The Science Teacher</i>. <b>64</b> (5): <span class="nowrap">16–</span>18. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24152064">24152064</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Science+Teacher&rft.atitle=The+Numbers+Game%3A+Playing+up+the+importance+of+significant+figures+and+scientific+notation&rft.volume=64&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E18&rft.date=1997&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24152064%23id-name%3DJSTOR&rft.aulast=Lustick&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMa2020" class="citation book cs1">Ma, Liping (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2Tr3DwAAQBAJ&pg=PA35"><i>Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-000-02734-1" title="Special:BookSources/978-1-000-02734-1"><bdi>978-1-000-02734-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knowing+and+Teaching+Elementary+Mathematics%3A+Teachers%27+Understanding+of+Fundamental+Mathematics+in+China+and+the+United+States&rft.pub=Routledge&rft.date=2020&rft.isbn=978-1-000-02734-1&rft.aulast=Ma&rft.aufirst=Liping&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2Tr3DwAAQBAJ%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaddenAubrey2017" class="citation book cs1">Madden, Daniel J.; Aubrey, Jason A. (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6EkzDwAAQBAJ&pg=PP18"><i>An Introduction to Proof Through Real Analysis</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-31472-1" title="Special:BookSources/978-1-119-31472-1"><bdi>978-1-119-31472-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Proof+Through+Real+Analysis&rft.pub=John+Wiley+%26+Sons&rft.date=2017&rft.isbn=978-1-119-31472-1&rft.aulast=Madden&rft.aufirst=Daniel+J.&rft.au=Aubrey%2C+Jason+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6EkzDwAAQBAJ%26pg%3DPP18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMahajan2010" class="citation book cs1">Mahajan, Sanjoy (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VrkZN0T0GaUC&pg=PA66"><i>Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-26559-1" title="Special:BookSources/978-0-262-26559-1"><bdi>978-0-262-26559-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Street-Fighting+Mathematics%3A+The+Art+of+Educated+Guessing+and+Opportunistic+Problem+Solving&rft.pub=MIT+Press&rft.date=2010&rft.isbn=978-0-262-26559-1&rft.aulast=Mahajan&rft.aufirst=Sanjoy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVrkZN0T0GaUC%26pg%3DPA66&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMarcusMcEvoy2016" class="citation book cs1">Marcus, Russell; McEvoy, Mark (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x29VCwAAQBAJ&pg=PA285"><i>An Historical Introduction to the Philosophy of Mathematics: A Reader</i></a>. Bloomsbury Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4725-3291-6" title="Special:BookSources/978-1-4725-3291-6"><bdi>978-1-4725-3291-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Historical+Introduction+to+the+Philosophy+of+Mathematics%3A+A+Reader&rft.pub=Bloomsbury+Publishing&rft.date=2016&rft.isbn=978-1-4725-3291-6&rft.aulast=Marcus&rft.aufirst=Russell&rft.au=McEvoy%2C+Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx29VCwAAQBAJ%26pg%3DPA285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMazumderEbong2023" class="citation book cs1">Mazumder, Pinaki; Ebong, Idongesit E. (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18"><i>Lectures on Digital Design Principles</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-000-92194-6" title="Special:BookSources/978-1-000-92194-6"><bdi>978-1-000-92194-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Digital+Design+Principles&rft.pub=CRC+Press&rft.date=2023&rft.isbn=978-1-000-92194-6&rft.aulast=Mazumder&rft.aufirst=Pinaki&rft.au=Ebong%2C+Idongesit+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7oS_EAAAQBAJ%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMazzolaMilmeisterWeissmann2004" class="citation book cs1">Mazzola, Guerino; Milmeister, Gérard; Weissmann, Jody (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66"><i>Comprehensive Mathematics For Computer Scientists 1: Sets And Numbers, Graphs And Algebra, Logic And Machines, Linear Geometry</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-20835-8" title="Special:BookSources/978-3-540-20835-8"><bdi>978-3-540-20835-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Comprehensive+Mathematics+For+Computer+Scientists+1%3A+Sets+And+Numbers%2C+Graphs+And+Algebra%2C+Logic+And+Machines%2C+Linear+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2004&rft.isbn=978-3-540-20835-8&rft.aulast=Mazzola&rft.aufirst=Guerino&rft.au=Milmeister%2C+G%C3%A9rard&rft.au=Weissmann%2C+Jody&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCkFCCA-2sRgC%26pg%3DPA66&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMeyer2023" class="citation book cs1">Meyer, Carl D. (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-X-_EAAAQBAJ&pg=PA234"><i>Matrix Analysis and Applied Linear Algebra: Second Edition</i></a>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61197-744-8" title="Special:BookSources/978-1-61197-744-8"><bdi>978-1-61197-744-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis+and+Applied+Linear+Algebra%3A+Second+Edition&rft.pub=SIAM&rft.date=2023&rft.isbn=978-1-61197-744-8&rft.aulast=Meyer&rft.aufirst=Carl+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-X-_EAAAQBAJ%26pg%3DPA234&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMonahan2012" class="citation book cs1">Monahan, John F. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aSv09LwmuRYC&pg=PA18">"2. Basic Computational Algorithms"</a>. In Gentle, James E.; Härdle, Wolfgang Karl; Mori, Yuichi (eds.). <i>Handbook of Computational Statistics: Concepts and Methods</i>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-21551-3" title="Special:BookSources/978-3-642-21551-3"><bdi>978-3-642-21551-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.+Basic+Computational+Algorithms&rft.btitle=Handbook+of+Computational+Statistics%3A+Concepts+and+Methods&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-3-642-21551-3&rft.aulast=Monahan&rft.aufirst=John+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaSv09LwmuRYC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMoncayo2018" class="citation book cs1">Moncayo, Raul (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J-pTDwAAQBAJ&pg=PT25"><i>Lalangue, Sinthome, Jouissance, and Nomination: A Reading Companion and Commentary on Lacan's Seminar XXIII on the Sinthome</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-91554-3" title="Special:BookSources/978-0-429-91554-3"><bdi>978-0-429-91554-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lalangue%2C+Sinthome%2C+Jouissance%2C+and+Nomination%3A+A+Reading+Companion+and+Commentary+on+Lacan%27s+Seminar+XXIII+on+the+Sinthome&rft.pub=Routledge&rft.date=2018&rft.isbn=978-0-429-91554-3&rft.aulast=Moncayo&rft.aufirst=Raul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJ-pTDwAAQBAJ%26pg%3DPT25&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMooneyBriggsHansenMcCullouch2014" class="citation book cs1">Mooney, Claire; Briggs, Mary; Hansen, Alice; McCullouch, Judith; Fletcher, Mike (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_dPgAwAAQBAJ&pg=PT148"><i>Primary Mathematics: Teaching Theory and Practice</i></a>. Learning Matters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4739-0707-2" title="Special:BookSources/978-1-4739-0707-2"><bdi>978-1-4739-0707-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primary+Mathematics%3A+Teaching+Theory+and+Practice&rft.pub=Learning+Matters&rft.date=2014&rft.isbn=978-1-4739-0707-2&rft.aulast=Mooney&rft.aufirst=Claire&rft.au=Briggs%2C+Mary&rft.au=Hansen%2C+Alice&rft.au=McCullouch%2C+Judith&rft.au=Fletcher%2C+Mike&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_dPgAwAAQBAJ%26pg%3DPT148&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMooreKearfottCloud2009" class="citation book cs1">Moore, Ramon E.; Kearfott, R. Baker; Cloud, Michael J. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kd8FmmN7sAoC&pg=PA10"><i>Introduction to Interval Analysis</i></a>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-669-6" title="Special:BookSources/978-0-89871-669-6"><bdi>978-0-89871-669-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Interval+Analysis&rft.pub=SIAM&rft.date=2009&rft.isbn=978-0-89871-669-6&rft.aulast=Moore&rft.aufirst=Ramon+E.&rft.au=Kearfott%2C+R.+Baker&rft.au=Cloud%2C+Michael+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkd8FmmN7sAoC%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMullerBrisebarreDinechinJeannerod2009" class="citation book cs1">Muller, Jean-Michel; Brisebarre, Nicolas; Dinechin, Florent de; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; <a href="/wiki/Nathalie_Revol" title="Nathalie Revol">Revol, Nathalie</a>; Stehlé, Damien; Torres, Serge (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=baFvrIOPvncC&pg=PA13"><i>Handbook of Floating-Point Arithmetic</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4705-6" title="Special:BookSources/978-0-8176-4705-6"><bdi>978-0-8176-4705-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Floating-Point+Arithmetic&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-0-8176-4705-6&rft.aulast=Muller&rft.aufirst=Jean-Michel&rft.au=Brisebarre%2C+Nicolas&rft.au=Dinechin%2C+Florent+de&rft.au=Jeannerod%2C+Claude-Pierre&rft.au=Lef%C3%A8vre%2C+Vincent&rft.au=Melquiond%2C+Guillaume&rft.au=Revol%2C+Nathalie&rft.au=Stehl%C3%A9%2C+Damien&rft.au=Torres%2C+Serge&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbaFvrIOPvncC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMullerBrunieDinechinJeannerod2018" class="citation book cs1">Muller, Jean-Michel; Brunie, Nicolas; Dinechin, Florent de; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume; <a href="/wiki/Nathalie_Revol" title="Nathalie Revol">Revol, Nathalie</a>; Torres, Serge (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h3ZZDwAAQBAJ&pg=PA539"><i>Handbook of Floating-Point Arithmetic</i></a>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-76526-6" title="Special:BookSources/978-3-319-76526-6"><bdi>978-3-319-76526-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Floating-Point+Arithmetic&rft.pub=Birkh%C3%A4user&rft.date=2018&rft.isbn=978-3-319-76526-6&rft.aulast=Muller&rft.aufirst=Jean-Michel&rft.au=Brunie%2C+Nicolas&rft.au=Dinechin%2C+Florent+de&rft.au=Jeannerod%2C+Claude-Pierre&rft.au=Joldes%2C+Mioara&rft.au=Lef%C3%A8vre%2C+Vincent&rft.au=Melquiond%2C+Guillaume&rft.au=Revol%2C+Nathalie&rft.au=Torres%2C+Serge&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh3ZZDwAAQBAJ%26pg%3DPA539&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMusserPetersonBurger2013" class="citation book cs1">Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA347"><i>Mathematics for Elementary Teachers: A Contemporary Approach</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-48700-6" title="Special:BookSources/978-1-118-48700-6"><bdi>978-1-118-48700-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Elementary+Teachers%3A+A+Contemporary+Approach&rft.pub=John+Wiley+%26+Sons&rft.date=2013&rft.isbn=978-1-118-48700-6&rft.aulast=Musser&rft.aufirst=Gary+L.&rft.au=Peterson%2C+Blake+E.&rft.au=Burger%2C+William+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8jh7DwAAQBAJ%26pg%3DPA347&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMW_staff2023" class="citation web cs1">MW staff (2023). <a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/arithmetic">"Definition of Arithmetic"</a>. <i>www.merriam-webster.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">19 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.merriam-webster.com&rft.atitle=Definition+of+Arithmetic&rft.date=2023&rft.au=MW+staff&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Farithmetic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNagel2002" class="citation book cs1">Nagel, Rob (2002). <a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic"><i>U-X-L Encyclopedia of Science</i></a>. U-X-L. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7876-5440-5" title="Special:BookSources/978-0-7876-5440-5"><bdi>978-0-7876-5440-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=U-X-L+Encyclopedia+of+Science&rft.pub=U-X-L&rft.date=2002&rft.isbn=978-0-7876-5440-5&rft.aulast=Nagel&rft.aufirst=Rob&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience-and-technology%2Fmathematics%2Fmathematics%2Farithmetic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNagelNewman2008" class="citation book cs1">Nagel, Ernest; Newman, James Roy (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4"><i>Godel's Proof</i></a>. NYU Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8147-5837-3" title="Special:BookSources/978-0-8147-5837-3"><bdi>978-0-8147-5837-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Godel%27s+Proof&rft.pub=NYU+Press&rft.date=2008&rft.isbn=978-0-8147-5837-3&rft.aulast=Nagel&rft.aufirst=Ernest&rft.au=Newman%2C+James+Roy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWgwUCgAAQBAJ%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNakovKolev2013" class="citation book cs1">Nakov, Svetlin; Kolev, Veselin (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA271"><i>Fundamentals of Computer Programming with C#: The Bulgarian C# Book</i></a>. Faber Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-954-400-773-7" title="Special:BookSources/978-954-400-773-7"><bdi>978-954-400-773-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Computer+Programming+with+C%23%3A+The+Bulgarian+C%23+Book&rft.pub=Faber+Publishing&rft.date=2013&rft.isbn=978-954-400-773-7&rft.aulast=Nakov&rft.aufirst=Svetlin&rft.au=Kolev%2C+Veselin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxYgCAQAAQBAJ%26pg%3DPA271&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNCTM_Staff" class="citation web cs1">NCTM Staff. <a rel="nofollow" class="external text" href="https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Number-and-Operations/">"Number and Operations"</a>. <i>www.nctm.org</i>. National Council of Teachers of Mathematics<span class="reference-accessdate">. Retrieved <span class="nowrap">21 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.nctm.org&rft.atitle=Number+and+Operations&rft.au=NCTM+Staff&rft_id=https%3A%2F%2Fwww.nctm.org%2FStandards-and-Positions%2FPrinciples-and-Standards%2FNumber-and-Operations%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNelson2019" class="citation book cs1">Nelson, Gerald (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xTiDDwAAQBAJ&pg=PR31"><i>English: An Essential Grammar</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-12273-3" title="Special:BookSources/978-1-351-12273-3"><bdi>978-1-351-12273-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=English%3A+An+Essential+Grammar&rft.pub=Routledge&rft.date=2019&rft.isbn=978-1-351-12273-3&rft.aulast=Nelson&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxTiDDwAAQBAJ%26pg%3DPR31&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNullLobur2006" class="citation book cs1">Null, Linda; Lobur, Julia (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QGPHAl9GE-IC&pg=PA40"><i>The Essentials of Computer Organization and Architecture</i></a>. Jones & Bartlett Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-3769-6" title="Special:BookSources/978-0-7637-3769-6"><bdi>978-0-7637-3769-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Essentials+of+Computer+Organization+and+Architecture&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2006&rft.isbn=978-0-7637-3769-6&rft.aulast=Null&rft.aufirst=Linda&rft.au=Lobur%2C+Julia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQGPHAl9GE-IC%26pg%3DPA40&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNurnberger-Haag2017" class="citation book cs1">Nurnberger-Haag, Julie (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Uls6DwAAQBAJ&pg=PA215">"Borrow, Trade, Regroup, or Unpack? Revealing How Instructional Metaphors Portray Base-Ten Number"</a>. In Jao, Limin; Radakovic, Nenad (eds.). <i>Transdisciplinarity in Mathematics Education: Blurring Disciplinary Boundaries</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-63624-5" title="Special:BookSources/978-3-319-63624-5"><bdi>978-3-319-63624-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Borrow%2C+Trade%2C+Regroup%2C+or+Unpack%3F+Revealing+How+Instructional+Metaphors+Portray+Base-Ten+Number&rft.btitle=Transdisciplinarity+in+Mathematics+Education%3A+Blurring+Disciplinary+Boundaries&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-63624-5&rft.aulast=Nurnberger-Haag&rft.aufirst=Julie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUls6DwAAQBAJ%26pg%3DPA215&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'Leary2015" class="citation book cs1">O'Leary, Michael L. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ci6kBgAAQBAJ&pg=PA190"><i>A First Course in Mathematical Logic and Set Theory</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-90588-3" title="Special:BookSources/978-0-470-90588-3"><bdi>978-0-470-90588-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Mathematical+Logic+and+Set+Theory&rft.pub=John+Wiley+%26+Sons&rft.date=2015&rft.isbn=978-0-470-90588-3&rft.aulast=O%27Leary&rft.aufirst=Michael+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCi6kBgAAQBAJ%26pg%3DPA190&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'Regan2012" class="citation book cs1">O'Regan, Gerard (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QqrItgm351EC&pg=PA24"><i>A Brief History of Computing</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-2359-0" title="Special:BookSources/978-1-4471-2359-0"><bdi>978-1-4471-2359-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brief+History+of+Computing&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4471-2359-0&rft.aulast=O%27Regan&rft.aufirst=Gerard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQqrItgm351EC%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOakes2020" class="citation book cs1">Oakes, Elizabeth (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RtyPEAAAQBAJ&pg=PA330"><i>Encyclopedia of World Scientists, Updated Edition</i></a>. Infobase Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4381-9545-2" title="Special:BookSources/978-1-4381-9545-2"><bdi>978-1-4381-9545-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+World+Scientists%2C+Updated+Edition&rft.pub=Infobase+Publishing&rft.date=2020&rft.isbn=978-1-4381-9545-2&rft.aulast=Oakes&rft.aufirst=Elizabeth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRtyPEAAAQBAJ%26pg%3DPA330&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOdomBarbarinWasik2009" class="citation book cs1">Odom, Samuel L.; Barbarin, Oscar A.; Wasik, Barbara Hanna (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MIuyhc8W-A8C&pg=PA589">"Applying Lessons from Developmental Science to Early Education"</a>. In Barbarin, Oscar A.; Wasik, Barbara Hanna (eds.). <i>Handbook of Child Development and Early Education: Research to Practice</i>. Guilford Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60623-302-3" title="Special:BookSources/978-1-60623-302-3"><bdi>978-1-60623-302-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Applying+Lessons+from+Developmental+Science+to+Early+Education&rft.btitle=Handbook+of+Child+Development+and+Early+Education%3A+Research+to+Practice&rft.pub=Guilford+Press&rft.date=2009&rft.isbn=978-1-60623-302-3&rft.aulast=Odom&rft.aufirst=Samuel+L.&rft.au=Barbarin%2C+Oscar+A.&rft.au=Wasik%2C+Barbara+Hanna&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMIuyhc8W-A8C%26pg%3DPA589&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOliver2005" class="citation book cs1">Oliver, Alexander D. (2005). "Arithmetic, Foundations of". In Honderich, Ted (ed.). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/HONTOC-2"><i>The Oxford Companion to Philosophy</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-926479-7" title="Special:BookSources/978-0-19-926479-7"><bdi>978-0-19-926479-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Arithmetic%2C+Foundations+of&rft.btitle=The+Oxford+Companion+to+Philosophy&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-926479-7&rft.aulast=Oliver&rft.aufirst=Alexander+D.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FHONTOC-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOmondi2020" class="citation book cs1">Omondi, Amos R. (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_m7NDwAAQBAJ&pg=PR8"><i>Cryptography Arithmetic: Algorithms and Hardware Architectures</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-34142-8" title="Special:BookSources/978-3-030-34142-8"><bdi>978-3-030-34142-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cryptography+Arithmetic%3A+Algorithms+and+Hardware+Architectures&rft.pub=Springer+Nature&rft.date=2020&rft.isbn=978-3-030-34142-8&rft.aulast=Omondi&rft.aufirst=Amos+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_m7NDwAAQBAJ%26pg%3DPR8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOngleyCarey2013" class="citation book cs1">Ongley, John; Carey, Rosalind (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PT24"><i>Russell: A Guide for the Perplexed</i></a>. Bloomsbury Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4411-9123-6" title="Special:BookSources/978-1-4411-9123-6"><bdi>978-1-4411-9123-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Russell%3A+A+Guide+for+the+Perplexed&rft.pub=Bloomsbury+Publishing&rft.date=2013&rft.isbn=978-1-4411-9123-6&rft.aulast=Ongley&rft.aufirst=John&rft.au=Carey%2C+Rosalind&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx-_KDwAAQBAJ%26pg%3DPT24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOre1948" class="citation book cs1">Ore, Oystein (1948). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Sl_6BPp7S0AC"><i>Number Theory and Its History</i></a>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65620-5" title="Special:BookSources/978-0-486-65620-5"><bdi>978-0-486-65620-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1397541">1397541</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+and+Its+History&rft.pub=McGraw-Hill&rft.date=1948&rft_id=info%3Aoclcnum%2F1397541&rft.isbn=978-0-486-65620-5&rft.aulast=Ore&rft.aufirst=Oystein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSl_6BPp7S0AC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOrr1995" class="citation book cs1">Orr, David B. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DMTqRnoE8iMC&pg=PA49"><i>Fundamentals of Applied Statistics and Surveys</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-98821-9" title="Special:BookSources/978-0-412-98821-9"><bdi>978-0-412-98821-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Applied+Statistics+and+Surveys&rft.pub=CRC+Press&rft.date=1995&rft.isbn=978-0-412-98821-9&rft.aulast=Orr&rft.aufirst=David+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDMTqRnoE8iMC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOtis2024" class="citation book cs1">Otis, Jessica Marie (2024). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=07jiEAAAQBAJ&pg=PA15"><i>By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-760877-7" title="Special:BookSources/978-0-19-760877-7"><bdi>978-0-19-760877-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=By+the+Numbers%3A+Numeracy%2C+Religion%2C+and+the+Quantitative+Transformation+of+Early+Modern+England&rft.pub=Oxford+University+Press&rft.date=2024&rft.isbn=978-0-19-760877-7&rft.aulast=Otis&rft.aufirst=Jessica+Marie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D07jiEAAAQBAJ%26pg%3DPA15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPaarPelzl2009" class="citation book cs1">Paar, Christof; Pelzl, Jan (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f24wFELSzkoC&pg=PA13"><i>Understanding Cryptography: A Textbook for Students and Practitioners</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-04101-3" title="Special:BookSources/978-3-642-04101-3"><bdi>978-3-642-04101-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Cryptography%3A+A+Textbook+for+Students+and+Practitioners&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-3-642-04101-3&rft.aulast=Paar&rft.aufirst=Christof&rft.au=Pelzl%2C+Jan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df24wFELSzkoC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPage2003" class="citation book cs1">Page, Robert L. (2003). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032">"Number Theory, Elementary"</a>. <i>Encyclopedia of Physical Science and Technology</i> (Third ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-227410-7" title="Special:BookSources/978-0-12-227410-7"><bdi>978-0-12-227410-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Number+Theory%2C+Elementary&rft.btitle=Encyclopedia+of+Physical+Science+and+Technology&rft.edition=Third&rft.pub=Academic+Press&rft.date=2003&rft.isbn=978-0-12-227410-7&rft.aulast=Page&rft.aufirst=Robert+L.&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fabs%2Fpii%2FB0122274105005032&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPayne2017" class="citation book cs1">Payne, Andrew (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qMU2DwAAQBAJ&pg=PA202"><i>The Teleology of Action in Plato's Republic</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-879902-3" title="Special:BookSources/978-0-19-879902-3"><bdi>978-0-19-879902-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Teleology+of+Action+in+Plato%27s+Republic&rft.pub=Oxford+University+Press&rft.date=2017&rft.isbn=978-0-19-879902-3&rft.aulast=Payne&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqMU2DwAAQBAJ%26pg%3DPA202&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeirce2015" class="citation book cs1">Peirce, Charles S. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oI_yCQAAQBAJ&pg=PA109"><i>Arithmetic</i></a>. Walter de Gruyter GmbH & Co KG. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-086970-5" title="Special:BookSources/978-3-11-086970-5"><bdi>978-3-11-086970-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetic&rft.pub=Walter+de+Gruyter+GmbH+%26+Co+KG&rft.date=2015&rft.isbn=978-3-11-086970-5&rft.aulast=Peirce&rft.aufirst=Charles+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoI_yCQAAQBAJ%26pg%3DPA109&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPharrJakobHumphreys2023" class="citation book cs1">Pharr, Matt; Jakob, Wenzel; Humphreys, Greg (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kUtwEAAAQBAJ&pg=PA1057"><i>Physically Based Rendering: From Theory to Implementation</i></a> (4 ed.). MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-37403-3" title="Special:BookSources/978-0-262-37403-3"><bdi>978-0-262-37403-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physically+Based+Rendering%3A+From+Theory+to+Implementation&rft.edition=4&rft.pub=MIT+Press&rft.date=2023&rft.isbn=978-0-262-37403-3&rft.aulast=Pharr&rft.aufirst=Matt&rft.au=Jakob%2C+Wenzel&rft.au=Humphreys%2C+Greg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkUtwEAAAQBAJ%26pg%3DPA1057&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPomerance2010" class="citation book cs1"><a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Pomerance, Carl</a> (2010). <a rel="nofollow" class="external text" href="https://math.dartmouth.edu/~carlp/PDF/pcm0049.pdf">"IV.3 Computational Number Theory"</a> <span class="cs1-format">(PDF)</span>. In <a href="/wiki/Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a>; <a href="/wiki/June_Barrow-Green" title="June Barrow-Green">Barrow-Green, June</a>; <a href="/wiki/Imre_Leader" title="Imre Leader">Leader, Imre</a> (eds.). <i>The Princeton Companion to Mathematics</i>. Princeton University Press. pp. <span class="nowrap">348–</span>362. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3039-8" title="Special:BookSources/978-1-4008-3039-8"><bdi>978-1-4008-3039-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=IV.3+Computational+Number+Theory&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E348-%3C%2Fspan%3E362&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=978-1-4008-3039-8&rft.aulast=Pomerance&rft.aufirst=Carl&rft_id=https%3A%2F%2Fmath.dartmouth.edu%2F~carlp%2FPDF%2Fpcm0049.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPomeranceSárközy1995" class="citation book cs1">Pomerance, C.; Sárközy, A. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ktBP5vUl5gC&pg=PA969">"Combinatorial Number Theory"</a>. In Graham, R. L. (ed.). <i>Handbook of Combinatorics</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-093384-9" title="Special:BookSources/978-0-08-093384-9"><bdi>978-0-08-093384-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Combinatorial+Number+Theory&rft.btitle=Handbook+of+Combinatorics&rft.pub=Elsevier&rft.date=1995&rft.isbn=978-0-08-093384-9&rft.aulast=Pomerance&rft.aufirst=C.&rft.au=S%C3%A1rk%C3%B6zy%2C+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5ktBP5vUl5gC%26pg%3DPA969&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPonticorvoSchmbriMiglino2019" class="citation book cs1">Ponticorvo, Michela; Schmbri, Massimiliano; Miglino, Orazio (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zSiXDwAAQBAJ&pg=PA33">"How to Improve Spatial and Numerical Cognition with a Game-Based and Technology-Enhanced Learning Approach"</a>. In Vicente, José Manuel Ferrández; Álvarez-Sánchez, José Ramón; López, Félix de la Paz; Moreo, Javier Toledo; Adeli, Hojjat (eds.). <i>Understanding the Brain Function and Emotions: 8th International Work-Conference on the Interplay Between Natural and Artificial Computation, IWINAC 2019, Almería, Spain, June 3–7, 2019, Proceedings, Part I</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-19591-5" title="Special:BookSources/978-3-030-19591-5"><bdi>978-3-030-19591-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=How+to+Improve+Spatial+and+Numerical+Cognition+with+a+Game-Based+and+Technology-Enhanced+Learning+Approach&rft.btitle=Understanding+the+Brain+Function+and+Emotions%3A+8th+International+Work-Conference+on+the+Interplay+Between+Natural+and+Artificial+Computation%2C+IWINAC+2019%2C+Almer%C3%ADa%2C+Spain%2C+June+3%E2%80%937%2C+2019%2C+Proceedings%2C+Part+I&rft.pub=Springer&rft.date=2019&rft.isbn=978-3-030-19591-5&rft.aulast=Ponticorvo&rft.aufirst=Michela&rft.au=Schmbri%2C+Massimiliano&rft.au=Miglino%2C+Orazio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzSiXDwAAQBAJ%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPrata2002" class="citation book cs1">Prata, Stephen (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MsizNs-zVMAC&pg=PA138"><i>C Primer Plus</i></a>. Sams Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-672-32222-8" title="Special:BookSources/978-0-672-32222-8"><bdi>978-0-672-32222-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C+Primer+Plus&rft.pub=Sams+Publishing&rft.date=2002&rft.isbn=978-0-672-32222-8&rft.aulast=Prata&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMsizNs-zVMAC%26pg%3DPA138&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFQuinteroRosario2016" class="citation book cs1">Quintero, Ana Helvia; Rosario, Hector (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YAXyCwAAQBAJ&pg=PA74"><i>Math Makes Sense!: A Constructivist Approach To The Teaching And Learning Of Mathematics</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-78326-866-5" title="Special:BookSources/978-1-78326-866-5"><bdi>978-1-78326-866-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Math+Makes+Sense%21%3A+A+Constructivist+Approach+To+The+Teaching+And+Learning+Of+Mathematics&rft.pub=World+Scientific&rft.date=2016&rft.isbn=978-1-78326-866-5&rft.aulast=Quintero&rft.aufirst=Ana+Helvia&rft.au=Rosario%2C+Hector&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYAXyCwAAQBAJ%26pg%3DPA74&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRajan2022" class="citation book cs1"><a href="/wiki/Hridesh_Rajan" title="Hridesh Rajan">Rajan, Hridesh</a> (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OCE6EAAAQBAJ&pg=PA17"><i>An Experiential Introduction to Principles of Programming Languages</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-36243-6" title="Special:BookSources/978-0-262-36243-6"><bdi>978-0-262-36243-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Experiential+Introduction+to+Principles+of+Programming+Languages&rft.pub=MIT+Press&rft.date=2022&rft.isbn=978-0-262-36243-6&rft.aulast=Rajan&rft.aufirst=Hridesh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOCE6EAAAQBAJ%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReilly2009" class="citation book cs1">Reilly, Norman R. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q33he4hOlKcC&pg=PA75"><i>Introduction to Applied Algebraic Systems</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-970992-2" title="Special:BookSources/978-0-19-970992-2"><bdi>978-0-19-970992-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Applied+Algebraic+Systems&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-970992-2&rft.aulast=Reilly&rft.aufirst=Norman+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq33he4hOlKcC%26pg%3DPA75&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReitano2010" class="citation book cs1">Reitano, Robert R. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JYX6AQAAQBAJ&pg=PA42"><i>Introduction to Quantitative Finance: A Math Tool Kit</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01369-7" title="Special:BookSources/978-0-262-01369-7"><bdi>978-0-262-01369-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantitative+Finance%3A+A+Math+Tool+Kit&rft.pub=MIT+Press&rft.date=2010&rft.isbn=978-0-262-01369-7&rft.aulast=Reitano&rft.aufirst=Robert+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJYX6AQAAQBAJ%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFResnickFord2012" class="citation book cs1">Resnick, L. B.; Ford, W. W. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xj-j8pw2HN8C&pg=PA110"><i>Psychology of Mathematics for Instruction</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-55759-0" title="Special:BookSources/978-1-136-55759-0"><bdi>978-1-136-55759-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Psychology+of+Mathematics+for+Instruction&rft.pub=Routledge&rft.date=2012&rft.isbn=978-1-136-55759-0&rft.aulast=Resnick&rft.aufirst=L.+B.&rft.au=Ford%2C+W.+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dxj-j8pw2HN8C%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReynolds2008" class="citation book cs1">Reynolds, Barbara E. (2008). "Abacus". In Selin, Helaine (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1"><i>Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-4559-2" title="Special:BookSources/978-1-4020-4559-2"><bdi>978-1-4020-4559-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abacus&rft.btitle=Encyclopaedia+of+the+History+of+Science%2C+Technology%2C+and+Medicine+in+Non-Western+Cultures&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008&rft.isbn=978-1-4020-4559-2&rft.aulast=Reynolds&rft.aufirst=Barbara+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkt9DIY1g9HYC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRiesel2012" class="citation book cs1">Riesel, Hans (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA2"><i>Prime Numbers and Computer Methods for Factorization</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-0251-6" title="Special:BookSources/978-1-4612-0251-6"><bdi>978-1-4612-0251-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers+and+Computer+Methods+for+Factorization&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-0251-6&rft.aulast=Riesel&rft.aufirst=Hans&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DITvaBwAAQBAJ%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRisingMatthewsSchoaffMatthew2021" class="citation book cs1">Rising, Gerald R.; Matthews, James R.; Schoaff, Eileen; Matthew, Judith (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hjVZEAAAQBAJ&pg=PA110"><i>About Mathematics</i></a>. Linus Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60797-892-3" title="Special:BookSources/978-1-60797-892-3"><bdi>978-1-60797-892-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=About+Mathematics&rft.pub=Linus+Learning&rft.date=2021&rft.isbn=978-1-60797-892-3&rft.aulast=Rising&rft.aufirst=Gerald+R.&rft.au=Matthews%2C+James+R.&rft.au=Schoaff%2C+Eileen&rft.au=Matthew%2C+Judith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhjVZEAAAQBAJ%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRobbins2006" class="citation book cs1">Robbins, Neville (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1"><i>Beginning Number Theory</i></a>. Jones & Bartlett Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-3768-9" title="Special:BookSources/978-0-7637-3768-9"><bdi>978-0-7637-3768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beginning+Number+Theory&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2006&rft.isbn=978-0-7637-3768-9&rft.aulast=Robbins&rft.aufirst=Neville&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTtLMrKDsDuIC%26pg%3DPR12-IA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRoddaLittle2015" class="citation book cs1">Rodda, Harvey J. E.; Little, Max A. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cb_dCgAAQBAJ&pg=PA7"><i>Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-based Approach</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-07659-9" title="Special:BookSources/978-1-119-07659-9"><bdi>978-1-119-07659-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Mathematical+and+Statistical+Techniques+in+Hydrology%3A+An+Examples-based+Approach&rft.pub=John+Wiley+%26+Sons&rft.date=2015&rft.isbn=978-1-119-07659-9&rft.aulast=Rodda&rft.aufirst=Harvey+J.+E.&rft.au=Little%2C+Max+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dcb_dCgAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRoedeForestJamshidi2018" class="citation book cs1">Roe, John; deForest, Russ; Jamshidi, Sara (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3ppYDwAAQBAJ&pg=PA24"><i>Mathematics for Sustainability</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-76660-7" title="Special:BookSources/978-3-319-76660-7"><bdi>978-3-319-76660-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Sustainability&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-319-76660-7&rft.aulast=Roe&rft.aufirst=John&rft.au=deForest%2C+Russ&rft.au=Jamshidi%2C+Sara&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3ppYDwAAQBAJ%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRomanowski2008" class="citation book cs1">Romanowski, Perry (2008). "Arithmetic". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). <a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic"><i>The Gale Encyclopedia of Science</i></a> (4th ed.). Thompson Gale. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4144-2877-2" title="Special:BookSources/978-1-4144-2877-2"><bdi>978-1-4144-2877-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Arithmetic&rft.btitle=The+Gale+Encyclopedia+of+Science&rft.edition=4th&rft.pub=Thompson+Gale&rft.date=2008&rft.isbn=978-1-4144-2877-2&rft.aulast=Romanowski&rft.aufirst=Perry&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience-and-technology%2Fmathematics%2Fmathematics%2Farithmetic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRooney2021" class="citation book cs1"><a href="/wiki/Anne_Rooney" title="Anne Rooney">Rooney, Anne</a> (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34"><i>Think Like a Mathematician</i></a>. The Rosen Publishing Group. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4994-7092-5" title="Special:BookSources/978-1-4994-7092-5"><bdi>978-1-4994-7092-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Think+Like+a+Mathematician&rft.pub=The+Rosen+Publishing+Group&rft.date=2021&rft.isbn=978-1-4994-7092-5&rft.aulast=Rooney&rft.aufirst=Anne&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWnVeEAAAQBAJ%26pg%3DPA34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRossi2011" class="citation book cs1">Rossi, Richard J. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kSwVGbBtel8C&pg=PA101"><i>Theorems, Corollaries, Lemmas, and Methods of Proof</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-03057-8" title="Special:BookSources/978-1-118-03057-8"><bdi>978-1-118-03057-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorems%2C+Corollaries%2C+Lemmas%2C+and+Methods+of+Proof&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-1-118-03057-8&rft.aulast=Rossi&rft.aufirst=Richard+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkSwVGbBtel8C%26pg%3DPA101&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRuthven2012" class="citation book cs1">Ruthven, Kenneth (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RjZzAgAAQBAJ&pg=PA435">"12. Calculators in the Mathematics Curriculum: The Scope of Personal Computational Technology"</a>. In Bishop, Alan; Clements, M. A. (Ken); Keitel-Kreidt, Christine; Kilpatrick, Jeremy; Laborde, Colette (eds.). <i>International Handbook of Mathematics Education</i>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-009-1465-0" title="Special:BookSources/978-94-009-1465-0"><bdi>978-94-009-1465-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=12.+Calculators+in+the+Mathematics+Curriculum%3A+The+Scope+of+Personal+Computational+Technology&rft.btitle=International+Handbook+of+Mathematics+Education&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-94-009-1465-0&rft.aulast=Ruthven&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRjZzAgAAQBAJ%26pg%3DPA435&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSallySally_(Jr.)2012" class="citation book cs1">Sally, Judith D.; Sally (Jr.), Paul J. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA3"><i>Integers, Fractions, and Arithmetic: A Guide for Teachers</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8798-1" title="Special:BookSources/978-0-8218-8798-1"><bdi>978-0-8218-8798-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integers%2C+Fractions%2C+and+Arithmetic%3A+A+Guide+for+Teachers&rft.pub=American+Mathematical+Soc.&rft.date=2012&rft.isbn=978-0-8218-8798-1&rft.aulast=Sally&rft.aufirst=Judith+D.&rft.au=Sally+%28Jr.%29%2C+Paul+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNtjq07-FA_IC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSeamanRosslerBurgin2023" class="citation book cs1">Seaman, William; Rossler, Otto E.; Burgin, Mark (2023). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=213PEAAAQBAJ&pg=PA226"><i>Chaos, Information, And The Future Of Physics: The Seaman-rossler Dialogue With Information Perspectives By Burgin And Seaman</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-7138-0" title="Special:BookSources/978-981-12-7138-0"><bdi>978-981-12-7138-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos%2C+Information%2C+And+The+Future+Of+Physics%3A+The+Seaman-rossler+Dialogue+With+Information+Perspectives+By+Burgin+And+Seaman&rft.pub=World+Scientific&rft.date=2023&rft.isbn=978-981-12-7138-0&rft.aulast=Seaman&rft.aufirst=William&rft.au=Rossler%2C+Otto+E.&rft.au=Burgin%2C+Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D213PEAAAQBAJ%26pg%3DPA226&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShiva2018" class="citation book cs1">Shiva, Sajjan G. (2018). <i>Introduction to Logic Design</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-98983-1" title="Special:BookSources/978-1-351-98983-1"><bdi>978-1-351-98983-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Logic+Design&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-351-98983-1&rft.aulast=Shiva&rft.aufirst=Sajjan+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSierpinskaLerman1996" class="citation book cs1">Sierpinska, Anna; Lerman, Stephen (1996). "Epistemologies of Mathematics and of Mathematics Education". In Alan J. Bishop; Ken Clements; Christine Keitel; Jeremy Kilpatrick; Colette Laborde (eds.). <i>International Handbook of Mathematics Education: Part 1</i>. Kluwer International Handbooks of Education, vol. 4. Springer Netherlands. pp. <span class="nowrap">827–</span>876. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-009-1465-0_23">10.1007/978-94-009-1465-0_23</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-009-1465-0" title="Special:BookSources/978-94-009-1465-0"><bdi>978-94-009-1465-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Epistemologies+of+Mathematics+and+of+Mathematics+Education&rft.btitle=International+Handbook+of+Mathematics+Education%3A+Part+1&rft.series=Kluwer+International+Handbooks+of+Education%2C+vol.+4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E827-%3C%2Fspan%3E876&rft.pub=Springer+Netherlands&rft.date=1996&rft_id=info%3Adoi%2F10.1007%2F978-94-009-1465-0_23&rft.isbn=978-94-009-1465-0&rft.aulast=Sierpinska&rft.aufirst=Anna&rft.au=Lerman%2C+Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1958" class="citation book cs1">Smith, David E. (1958). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC&pg=PA7"><i>History of Mathematics</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-20430-7" title="Special:BookSources/978-0-486-20430-7"><bdi>978-0-486-20430-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pub=Courier+Corporation&rft.date=1958&rft.isbn=978-0-486-20430-7&rft.aulast=Smith&rft.aufirst=David+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuTytJGnTf1kC%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmyth1864" class="citation book cs1">Smyth, William (1864). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BqQZAAAAYAAJ&pg=PA55"><i>Elementary Algebra: For Schools and Academies</i></a>. Bailey and Noyes. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/3901163143">3901163143</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Algebra%3A+For+Schools+and+Academies&rft.pub=Bailey+and+Noyes&rft.date=1864&rft_id=info%3Aoclcnum%2F3901163143&rft.aulast=Smyth&rft.aufirst=William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBqQZAAAAYAAJ%26pg%3DPA55&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSophian2017" class="citation book cs1">Sophian, Catherine (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8FYPEAAAQBAJ&pg=PA84"><i>The Origins of Mathematical Knowledge in Childhood</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-54175-6" title="Special:BookSources/978-1-351-54175-6"><bdi>978-1-351-54175-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Origins+of+Mathematical+Knowledge+in+Childhood&rft.pub=Routledge&rft.date=2017&rft.isbn=978-1-351-54175-6&rft.aulast=Sophian&rft.aufirst=Catherine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8FYPEAAAQBAJ%26pg%3DPA84&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSperlingStuart1981" class="citation book cs1">Sperling, Abraham; Stuart, Monroe (1981). <i>Mathematics</i>. Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-0405-5" title="Special:BookSources/978-0-7506-0405-5"><bdi>978-0-7506-0405-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics&rft.pub=Elsevier+Science&rft.date=1981&rft.isbn=978-0-7506-0405-5&rft.aulast=Sperling&rft.aufirst=Abraham&rft.au=Stuart%2C+Monroe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStakhov2020" class="citation book cs1">Stakhov, Alexey (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA73"><i>Mathematics Of Harmony As A New Interdisciplinary Direction And 'Golden' Paradigm Of Modern Science - Volume 2: Algorithmic Measurement Theory, Fibonacci And Golden Arithmetic's And Ternary Mirror-symmetrical Arithmetic</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-1348-9" title="Special:BookSources/978-981-12-1348-9"><bdi>978-981-12-1348-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Of+Harmony+As+A+New+Interdisciplinary+Direction+And+%27Golden%27+Paradigm+Of+Modern+Science+-+Volume+2%3A+Algorithmic+Measurement+Theory%2C+Fibonacci+And+Golden+Arithmetic%27s+And+Ternary+Mirror-symmetrical+Arithmetic&rft.pub=World+Scientific&rft.date=2020&rft.isbn=978-981-12-1348-9&rft.aulast=Stakhov&rft.aufirst=Alexey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFkn9DwAAQBAJ%26pg%3DPA73&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSternbergBen-Zeev2012" class="citation book cs1">Sternberg, Robert J.; Ben-Zeev, Talia (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q7F777rDl1AC&pg=PA95"><i>The Nature of Mathematical Thinking</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-48750-7" title="Special:BookSources/978-1-136-48750-7"><bdi>978-1-136-48750-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+of+Mathematical+Thinking&rft.pub=Routledge&rft.date=2012&rft.isbn=978-1-136-48750-7&rft.aulast=Sternberg&rft.aufirst=Robert+J.&rft.au=Ben-Zeev%2C+Talia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq7F777rDl1AC%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStevensonWaite2011" class="citation book cs1">Stevenson, Angus; Waite, Maurice (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sYScAQAAQBAJ&pg=PA70"><i>Concise Oxford English Dictionary: Luxury Edition</i></a>. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-960111-0" title="Special:BookSources/978-0-19-960111-0"><bdi>978-0-19-960111-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concise+Oxford+English+Dictionary%3A+Luxury+Edition&rft.pub=OUP+Oxford&rft.date=2011&rft.isbn=978-0-19-960111-0&rft.aulast=Stevenson&rft.aufirst=Angus&rft.au=Waite%2C+Maurice&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsYScAQAAQBAJ%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStewart2022" class="citation book cs1">Stewart, David E. (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=twafEAAAQBAJ&pg=PA26"><i>Numerical Analysis: A Graduate Course</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-08121-7" title="Special:BookSources/978-3-031-08121-7"><bdi>978-3-031-08121-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Analysis%3A+A+Graduate+Course&rft.pub=Springer+Nature&rft.date=2022&rft.isbn=978-3-031-08121-7&rft.aulast=Stewart&rft.aufirst=David+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtwafEAAAQBAJ%26pg%3DPA26&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStrathern2012" class="citation book cs1">Strathern, Paul (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NJm7Z5iQE24C&pg=PP9"><i>Turing And The Computer</i></a>. Random House. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4481-0656-1" title="Special:BookSources/978-1-4481-0656-1"><bdi>978-1-4481-0656-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Turing+And+The+Computer&rft.pub=Random+House&rft.date=2012&rft.isbn=978-1-4481-0656-1&rft.aulast=Strathern&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNJm7Z5iQE24C%26pg%3DPP9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSwanson2021" class="citation book cs1">Swanson, Irena (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cHshEAAAQBAJ&pg=PA107"><i>Introduction To Analysis With Complex Numbers</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-2587-1" title="Special:BookSources/978-981-12-2587-1"><bdi>978-981-12-2587-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Analysis+With+Complex+Numbers&rft.pub=World+Scientific&rft.date=2021&rft.isbn=978-981-12-2587-1&rft.aulast=Swanson&rft.aufirst=Irena&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcHshEAAAQBAJ%26pg%3DPA107&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSwartzlander2017" class="citation book cs1">Swartzlander, Earl E. (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VOnyWUUUj04C&pg=SA11-PA19">"High-Speed Computer Arithmetic"</a>. In Oklobdzija, Vojin G. (ed.). <i>Digital Design and Fabrication</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-8604-6" title="Special:BookSources/978-0-8493-8604-6"><bdi>978-0-8493-8604-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=High-Speed+Computer+Arithmetic&rft.btitle=Digital+Design+and+Fabrication&rft.pub=CRC+Press&rft.date=2017&rft.isbn=978-0-8493-8604-6&rft.aulast=Swartzlander&rft.aufirst=Earl+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVOnyWUUUj04C%26pg%3DSA11-PA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTarasov2008" class="citation book cs1">Tarasov, Vasily (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pHK11tfdE3QC&pg=PA57"><i>Quantum Mechanics of Non-Hamiltonian and Dissipative Systems</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-055971-1" title="Special:BookSources/978-0-08-055971-1"><bdi>978-0-08-055971-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+of+Non-Hamiltonian+and+Dissipative+Systems&rft.pub=Elsevier&rft.date=2008&rft.isbn=978-0-08-055971-1&rft.aulast=Tarasov&rft.aufirst=Vasily&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpHK11tfdE3QC%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaylor2012" class="citation book cs1">Taylor, Joseph L. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8"><i>Foundations of Analysis</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8984-8" title="Special:BookSources/978-0-8218-8984-8"><bdi>978-0-8218-8984-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Analysis&rft.pub=American+Mathematical+Soc.&rft.date=2012&rft.isbn=978-0-8218-8984-8&rft.aulast=Taylor&rft.aufirst=Joseph+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0ky2W1loV-QC%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThiamRochon2019" class="citation book cs1">Thiam, Thierno; Rochon, Gilbert (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164"><i>Sustainability, Emerging Technologies, and Pan-Africanism</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-22180-5" title="Special:BookSources/978-3-030-22180-5"><bdi>978-3-030-22180-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sustainability%2C+Emerging+Technologies%2C+and+Pan-Africanism&rft.pub=Springer+Nature&rft.date=2019&rft.isbn=978-3-030-22180-5&rft.aulast=Thiam&rft.aufirst=Thierno&rft.au=Rochon%2C+Gilbert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEWSsDwAAQBAJ%26pg%3DPA164&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTiles2009" class="citation book cs1">Tiles, Mary (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mbn35b2ghgkC&pg=PA243">"A Kantian Perspective on the Philosophy of Mathematics"</a>. In Irvine, Andrew D. (ed.). <i>Philosophy of Mathematics</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-093058-9" title="Special:BookSources/978-0-08-093058-9"><bdi>978-0-08-093058-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Kantian+Perspective+on+the+Philosophy+of+Mathematics&rft.btitle=Philosophy+of+Mathematics&rft.pub=Elsevier&rft.date=2009&rft.isbn=978-0-08-093058-9&rft.aulast=Tiles&rft.aufirst=Mary&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dmbn35b2ghgkC%26pg%3DPA243&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFUspenskiiSemenov2001" class="citation book cs1">Uspenskii, V. A.; Semenov, A. L. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yID37VIW-t8C&pg=PA113">"Solvable and Unsolvable Algorithmic Problems"</a>. In Tabachnikov, Serge (ed.). <i>Kvant Selecta: Combinatorics, I: Combinatorics, I</i>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2171-8" title="Special:BookSources/978-0-8218-2171-8"><bdi>978-0-8218-2171-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Solvable+and+Unsolvable+Algorithmic+Problems&rft.btitle=Kvant+Selecta%3A+Combinatorics%2C+I%3A+Combinatorics%2C+I&rft.pub=American+Mathematical+Soc.&rft.date=2001&rft.isbn=978-0-8218-2171-8&rft.aulast=Uspenskii&rft.aufirst=V.+A.&rft.au=Semenov%2C+A.+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyID37VIW-t8C%26pg%3DPA113&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVaccaroPepiciello2022" class="citation book cs1">Vaccaro, Alfredo; Pepiciello, Antonio (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tZxBEAAAQBAJ&pg=PA9"><i>Affine Arithmetic-Based Methods for Uncertain Power System Analysis</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-323-90503-9" title="Special:BookSources/978-0-323-90503-9"><bdi>978-0-323-90503-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Affine+Arithmetic-Based+Methods+for+Uncertain+Power+System+Analysis&rft.pub=Elsevier&rft.date=2022&rft.isbn=978-0-323-90503-9&rft.aulast=Vaccaro&rft.aufirst=Alfredo&rft.au=Pepiciello%2C+Antonio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtZxBEAAAQBAJ%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVerschaffelTorbeynsDe_Smedt2011" class="citation book cs1">Verschaffel, Lieven; Torbeyns, Joke; De Smedt, Bert (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177">"Mental Arithmetic"</a>. In Seel, Norbert M. (ed.). <i>Encyclopedia of the Sciences of Learning</i>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-1427-9" title="Special:BookSources/978-1-4419-1427-9"><bdi>978-1-4419-1427-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mental+Arithmetic&rft.btitle=Encyclopedia+of+the+Sciences+of+Learning&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-1-4419-1427-9&rft.aulast=Verschaffel&rft.aufirst=Lieven&rft.au=Torbeyns%2C+Joke&rft.au=De+Smedt%2C+Bert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxZuSxo4JxoAC%26pg%3DPA2177&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVictoria_Department_of_Education_Staff2023" class="citation web cs1">Victoria Department of Education Staff (2023). <a rel="nofollow" class="external text" href="https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/Pages/numeracy-for-all-learners.aspx">"Numeracy for All Learners"</a>. <i>www.education.vic.gov.au</i><span class="reference-accessdate">. Retrieved <span class="nowrap">22 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.education.vic.gov.au&rft.atitle=Numeracy+for+All+Learners&rft.date=2023&rft.au=Victoria+Department+of+Education+Staff&rft_id=https%3A%2F%2Fwww.education.vic.gov.au%2Fschool%2Fteachers%2Fteachingresources%2Fdiscipline%2Fmaths%2FPages%2Fnumeracy-for-all-learners.aspx&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVinogradov2019" class="citation web cs1">Vinogradov, A. I. (2019). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Algebraic_number_theory">"Algebraic Number Theory"</a>. <i>Encyclopedia of Mathematics</i>. Springer<span class="reference-accessdate">. Retrieved <span class="nowrap">23 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Algebraic+Number+Theory&rft.date=2019&rft.aulast=Vinogradov&rft.aufirst=A.+I.&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FAlgebraic_number_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVullo2020" class="citation book cs1">Vullo, Vincenzo (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g67QDwAAQBAJ&pg=PA140"><i>Gears: Volume 3: A Concise History</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-40164-1" title="Special:BookSources/978-3-030-40164-1"><bdi>978-3-030-40164-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gears%3A+Volume+3%3A+A+Concise+History&rft.pub=Springer+Nature&rft.date=2020&rft.isbn=978-3-030-40164-1&rft.aulast=Vullo&rft.aufirst=Vincenzo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg67QDwAAQBAJ%26pg%3DPA140&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWaite2013" class="citation book cs1">Waite, Maurice (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xqKcAQAAQBAJ&pg=PA42"><i>Pocket Oxford English Dictionary</i></a>. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-966615-7" title="Special:BookSources/978-0-19-966615-7"><bdi>978-0-19-966615-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pocket+Oxford+English+Dictionary&rft.pub=OUP+Oxford&rft.date=2013&rft.isbn=978-0-19-966615-7&rft.aulast=Waite&rft.aufirst=Maurice&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxqKcAQAAQBAJ%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWallis2011" class="citation book cs1">Wallis, W. D. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=18W4_LJ5bL0C&pg=PA303"><i>A Beginner's Guide to Discrete Mathematics</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-8286-6" title="Special:BookSources/978-0-8176-8286-6"><bdi>978-0-8176-8286-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Beginner%27s+Guide+to+Discrete+Mathematics&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-0-8176-8286-6&rft.aulast=Wallis&rft.aufirst=W.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D18W4_LJ5bL0C%26pg%3DPA303&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWallis2013" class="citation book cs1">Wallis, W. D. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ONgRBwAAQBAJ&pg=PA20"><i>A Beginner's Guide to Discrete Mathematics</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-3826-1" title="Special:BookSources/978-1-4757-3826-1"><bdi>978-1-4757-3826-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Beginner%27s+Guide+to+Discrete+Mathematics&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4757-3826-1&rft.aulast=Wallis&rft.aufirst=W.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DONgRBwAAQBAJ%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWang1997" class="citation book cs1">Wang, Hao (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pckvCy6L_ocC&pg=PA334"><i>A Logical Journey: From Gödel to Philosophy</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-26125-8" title="Special:BookSources/978-0-262-26125-8"><bdi>978-0-262-26125-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Logical+Journey%3A+From+G%C3%B6del+to+Philosophy&rft.pub=MIT+Press&rft.date=1997&rft.isbn=978-0-262-26125-8&rft.aulast=Wang&rft.aufirst=Hao&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpckvCy6L_ocC%26pg%3DPA334&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWard2012" class="citation book cs1">Ward, J. P. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LVDvCAAAQBAJ&pg=PA55"><i>Quaternions and Cayley Numbers: Algebra and Applications</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-011-5768-1" title="Special:BookSources/978-94-011-5768-1"><bdi>978-94-011-5768-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quaternions+and+Cayley+Numbers%3A+Algebra+and+Applications&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-94-011-5768-1&rft.aulast=Ward&rft.aufirst=J.+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLVDvCAAAQBAJ%26pg%3DPA55&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWedell2015" class="citation book cs1">Wedell, Moritz (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Uh5pCgAAQBAJ&pg=PA1235">"Numbers"</a>. In Classen, Albrecht (ed.). <i>Handbook of Medieval Culture. Volume 2</i>. Walter de Gruyter GmbH & Co KG. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-037763-7" title="Special:BookSources/978-3-11-037763-7"><bdi>978-3-11-037763-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Numbers&rft.btitle=Handbook+of+Medieval+Culture.+Volume+2&rft.pub=Walter+de+Gruyter+GmbH+%26+Co+KG&rft.date=2015&rft.isbn=978-3-11-037763-7&rft.aulast=Wedell&rft.aufirst=Moritz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUh5pCgAAQBAJ%26pg%3DPA1235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeil2009" class="citation book cs1">Weil, André (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ar7gBwAAQBAJ&pg=PR9"><i>Number Theory: An Approach Through History From Hammurapi to Legendre</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4571-7" title="Special:BookSources/978-0-8176-4571-7"><bdi>978-0-8176-4571-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory%3A+An+Approach+Through+History+From+Hammurapi+to+Legendre&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-0-8176-4571-7&rft.aulast=Weil&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAr7gBwAAQBAJ%26pg%3DPR9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeir2022" class="citation web cs1">Weir, Alan (2022). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/formalism-mathematics/">"Formalism in the Philosophy of Mathematics"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. Metaphysics Research Lab, Stanford University<span class="reference-accessdate">. Retrieved <span class="nowrap">22 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Formalism+in+the+Philosophy+of+Mathematics&rft.date=2022&rft.aulast=Weir&rft.aufirst=Alan&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fformalism-mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWheater2015" class="citation book cs1">Wheater, Carolyn (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19"><i>Algebra I</i></a>. Dorling Kindersley Limited. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-241-88779-0" title="Special:BookSources/978-0-241-88779-0"><bdi>978-0-241-88779-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+I&rft.pub=Dorling+Kindersley+Limited&rft.date=2015&rft.isbn=978-0-241-88779-0&rft.aulast=Wheater&rft.aufirst=Carolyn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQ7R3EAAAQBAJ%26pg%3DPP19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilson2020" class="citation book cs1">Wilson, Robin (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1"><i>Number Theory: A Very Short Introduction</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-879809-5" title="Special:BookSources/978-0-19-879809-5"><bdi>978-0-19-879809-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory%3A+A+Very+Short+Introduction&rft.pub=Oxford+University+Press&rft.date=2020&rft.isbn=978-0-19-879809-5&rft.aulast=Wilson&rft.aufirst=Robin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfcDgDwAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWrightEllemor-CollinsTabor2011" class="citation book cs1">Wright, Robert J.; Ellemor-Collins, David; Tabor, Pamela D. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136"><i>Developing Number Knowledge: Assessment, Teaching and Intervention with 7-11 Year Olds</i></a>. SAGE. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4462-8927-3" title="Special:BookSources/978-1-4462-8927-3"><bdi>978-1-4462-8927-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Developing+Number+Knowledge%3A+Assessment%2C+Teaching+and+Intervention+with+7-11+Year+Olds&rft.pub=SAGE&rft.date=2011&rft.isbn=978-1-4462-8927-3&rft.aulast=Wright&rft.aufirst=Robert+J.&rft.au=Ellemor-Collins%2C+David&rft.au=Tabor%2C+Pamela+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3yqdEAAAQBAJ%26pg%3DPA136&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFXuZhang2022" class="citation book cs1">Xu, Zhiwei; Zhang, Jialin (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s2RXEAAAQBAJ&pg=PA121"><i>Computational Thinking: A Perspective on Computer Science</i></a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-16-3848-0" title="Special:BookSources/978-981-16-3848-0"><bdi>978-981-16-3848-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Thinking%3A+A+Perspective+on+Computer+Science&rft.pub=Springer+Nature&rft.date=2022&rft.isbn=978-981-16-3848-0&rft.aulast=Xu&rft.aufirst=Zhiwei&rft.au=Zhang%2C+Jialin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ds2RXEAAAQBAJ%26pg%3DPA121&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYadin2016" class="citation book cs1">Yadin, Aharon (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KzeLDQAAQBAJ&pg=PT24"><i>Computer Systems Architecture</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-315-35592-4" title="Special:BookSources/978-1-315-35592-4"><bdi>978-1-315-35592-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Systems+Architecture&rft.pub=CRC+Press&rft.date=2016&rft.isbn=978-1-315-35592-4&rft.aulast=Yadin&rft.aufirst=Aharon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKzeLDQAAQBAJ%26pg%3DPT24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYan2002" class="citation book cs1">Yan, Song Y. (2002). <i>Number Theory for Computing</i>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-07710-4" title="Special:BookSources/978-3-642-07710-4"><bdi>978-3-642-07710-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+for+Computing&rft.pub=Springer+Science+%26+Business+Media&rft.date=2002&rft.isbn=978-3-642-07710-4&rft.aulast=Yan&rft.aufirst=Song+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYan2013a" class="citation book cs1">Yan, Song Y. (2013a). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15"><i>Computational Number Theory and Modern Cryptography</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-18858-3" title="Special:BookSources/978-1-118-18858-3"><bdi>978-1-118-18858-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Number+Theory+and+Modern+Cryptography&rft.pub=John+Wiley+%26+Sons&rft.date=2013&rft.isbn=978-1-118-18858-3&rft.aulast=Yan&rft.aufirst=Song+Y.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D74oBi4ys0UUC%26pg%3DPA15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoung2010" class="citation book cs1"><a href="/wiki/Cynthia_Y._Young" title="Cynthia Y. Young">Young, Cynthia Y.</a> (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA994"><i>Precalculus</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-75684-2" title="Special:BookSources/978-0-471-75684-2"><bdi>978-0-471-75684-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus&rft.pub=John+Wiley+%26+Sons&rft.date=2010&rft.isbn=978-0-471-75684-2&rft.aulast=Young&rft.aufirst=Cynthia+Y.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9HRLAn326zEC%26pg%3DRA1-PA994&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoung2021" class="citation book cs1">Young, Cynthia Y. (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hpVFEAAAQBAJ&pg=RA1-PA4"><i>Algebra and Trigonometry</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-77830-1" title="Special:BookSources/978-1-119-77830-1"><bdi>978-1-119-77830-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+and+Trigonometry&rft.pub=John+Wiley+%26+Sons&rft.date=2021&rft.isbn=978-1-119-77830-1&rft.aulast=Young&rft.aufirst=Cynthia+Y.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhpVFEAAAQBAJ%26pg%3DRA1-PA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZhang2012" class="citation book cs1">Zhang, G. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GRTSBwAAQBAJ&pg=PA130"><i>Logic of Domains</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-0445-9" title="Special:BookSources/978-1-4612-0445-9"><bdi>978-1-4612-0445-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+of+Domains&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4612-0445-9&rft.aulast=Zhang&rft.aufirst=G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGRTSBwAAQBAJ%26pg%3DPA130&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArithmetic" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arithmetic&action=edit&section=29" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236088147">.mw-parser-output .sister-bar{display:flex;justify-content:center;align-items:baseline;font-size:88%;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em 0 0;padding:0 2em}.mw-parser-output .sister-bar-header{margin:0 1em 0 0.5em;padding:0.2em 0;flex:0 0 auto;min-height:24px;line-height:22px}.mw-parser-output .sister-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;align-items:baseline;padding:0.2em 0;column-gap:1em;margin:0;list-style:none}.mw-parser-output .sister-bar-item{display:flex;align-items:baseline;margin:0.15em 0;min-height:24px;text-align:left}.mw-parser-output .sister-bar-logo{width:22px;line-height:22px;margin:0 0.2em;text-align:right}.mw-parser-output .sister-bar-link{margin:0 0.2em;text-align:left}@media screen and (max-width:960px){.mw-parser-output .sister-bar{flex-flow:column wrap;margin:1em auto 0}.mw-parser-output .sister-bar-header{flex:0 1}.mw-parser-output .sister-bar-content{flex:1;border-top:1px solid #a2a9b1;margin:0;list-style:none}.mw-parser-output .sister-bar-item{flex:0 0 20em;min-width:20em}}.mw-parser-output .navbox+link+.sister-bar,.mw-parser-output .navbox+style+.sister-bar,.mw-parser-output .portal-bar+link+.sister-bar,.mw-parser-output .portal-bar+style+.sister-bar,.mw-parser-output .sister-bar+.navbox-styles+.navbox,.mw-parser-output .sister-bar+.navbox-styles+.portal-bar{margin-top:-1px}@media print{body.ns-0 .mw-parser-output .sister-bar{display:none!important}}</style><div class="noprint metadata sister-bar" role="navigation" aria-label="sister-projects"><div class="sister-bar-header"><b>Arithmetic</b> at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects"><span id="sister-projects" style="white-space:nowrap;">sister projects</span></a>:</div><ul class="sister-bar-content"><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/19px-Wiktionary-logo-v2.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/29px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/38px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /></span></span></span><span class="sister-bar-link"><b><a href="https://en.wiktionary.org/wiki/Special:Search/Arithmetic" class="extiw" title="wikt:Special:Search/Arithmetic">Definitions</a></b> from Wiktionary</span></li><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/14px-Commons-logo.svg.png" decoding="async" width="14" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/21px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/28px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-bar-link"><b><a href="https://commons.wikimedia.org/wiki/Arithmetic" class="extiw" title="c:Arithmetic">Media</a></b> from Commons</span></li><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/16px-Wikiquote-logo.svg.png" decoding="async" width="16" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/24px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/32px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></span><span class="sister-bar-link"><b><a href="https://en.wikiquote.org/wiki/Arithmetic" class="extiw" title="q:Arithmetic">Quotations</a></b> from Wikiquote</span></li><li class="sister-bar-item"><span class="sister-bar-logo"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/19px-Wikibooks-logo.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/29px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/38px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></span><span class="sister-bar-link"><b><a href="https://en.wikibooks.org/wiki/Arithmetic" class="extiw" title="b:Arithmetic">Textbooks</a></b> from Wikibooks</span></li></ul></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas1051" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas1051" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q11205#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1219" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q11205#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1219" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11205#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4002919-0">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Arithmetic"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85007163">United States</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00570203">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="aritmetika"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph118605&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007294694905171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://esu.com.ua/search_articles.php?id=43227">Encyclopedia of Modern Ukraine</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Arithmetic&oldid=1272376224">https://en.wikipedia.org/w/index.php?title=Arithmetic&oldid=1272376224</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Arithmetic" title="Category:Arithmetic">Arithmetic</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Good_articles" title="Category:Good articles">Good articles</a></li><li><a href="/wiki/Category:Pages_using_multiple_image_with_auto_scaled_images" title="Category:Pages using multiple image with auto scaled images">Pages using multiple image with auto scaled images</a></li><li><a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_December_2024" title="Category:CS1 maint: DOI inactive as of December 2024">CS1 maint: DOI inactive as of December 2024</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 January 2025, at 09:16<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Arithmetic&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Arithmetic</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>163 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-76d4c66f66-482fg","wgBackendResponseTime":366,"wgPageParseReport":{"limitreport":{"cputime":"2.537","walltime":"2.912","ppvisitednodes":{"value":20574,"limit":1000000},"postexpandincludesize":{"value":677075,"limit":2097152},"templateargumentsize":{"value":3447,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":27,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":735978,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 2209.147 1 -total"," 39.33% 868.840 190 Template:Cite_book"," 24.63% 544.116 2 Template:Reflist"," 19.36% 427.658 164 Template:Multiref"," 13.30% 293.801 463 Template:Harvnb"," 5.22% 115.267 10 Template:Efn"," 4.77% 105.369 1 Template:Sfn"," 4.11% 90.695 1 Template:Areas_of_mathematics"," 3.98% 87.873 1 Template:Navbox"," 3.84% 84.930 1 Template:Short_description"]},"scribunto":{"limitreport-timeusage":{"value":"1.597","limit":"10.000"},"limitreport-memusage":{"value":17434131,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAchatzAnderson2005\"] = 1,\n [\"CITEREFAdamowicz1994\"] = 1,\n [\"CITEREFAdhamiMeenenMeenenHite2007\"] = 1,\n [\"CITEREFAli_RahmanShahrillAbbasTan2017\"] = 1,\n [\"CITEREFAngLam2004\"] = 1,\n [\"CITEREFAsano2013\"] = 1,\n [\"CITEREFAskew2010\"] = 1,\n [\"CITEREFAubrey1999\"] = 1,\n [\"CITEREFBagaria2023\"] = 1,\n [\"CITEREFBarnesRiceHanoch2017\"] = 1,\n [\"CITEREFBerchGearyKoepke2015\"] = 1,\n [\"CITEREFBird2021\"] = 1,\n [\"CITEREFBloch2011\"] = 1,\n [\"CITEREFBohacek2009\"] = 1,\n [\"CITEREFBookerBondSparrowSwan2015\"] = 1,\n [\"CITEREFBradley2006\"] = 1,\n [\"CITEREFBrentZimmermann2010\"] = 1,\n [\"CITEREFBronshteinSemendyayevMusiolMühlig2015\"] = 1,\n [\"CITEREFBrown2010\"] = 1,\n [\"CITEREFBruderer2021\"] = 1,\n [\"CITEREFBuddSangwin2001\"] = 1,\n [\"CITEREFBukhshtabNechaev2014\"] = 1,\n [\"CITEREFBukhshtabNechaev2016\"] = 1,\n [\"CITEREFBukhshtabPechaev2020\"] = 1,\n [\"CITEREFBurgin2022\"] = 1,\n [\"CITEREFCafaroEpicocoPulimeno2018\"] = 1,\n [\"CITEREFCai2023\"] = 1,\n [\"CITEREFCampbell-KellyCroarkenFloodRobson2007\"] = 1,\n [\"CITEREFCampbell2012\"] = 1,\n [\"CITEREFCaprioAveniMukherjee2022\"] = 1,\n [\"CITEREFCarraherSchliemann2015\"] = 1,\n [\"CITEREFCavanagh2017\"] = 1,\n [\"CITEREFChakravertyRout2022\"] = 1,\n [\"CITEREFChemlaKellerProust2023\"] = 1,\n [\"CITEREFCignoniCossu2016\"] = 1,\n [\"CITEREFCohen2003\"] = 1,\n [\"CITEREFColyvan2023\"] = 1,\n [\"CITEREFConfrey1994\"] = 1,\n [\"CITEREFConradieGoranko2015\"] = 1,\n [\"CITEREFCryer2014\"] = 1,\n [\"CITEREFCunningham2016\"] = 1,\n [\"CITEREFCurley2011\"] = 1,\n [\"CITEREFCuytPetersenVerdonkWaadeland2008\"] = 1,\n [\"CITEREFDavisGouldingSuggate2017\"] = 1,\n [\"CITEREFDe_CruzNethSchlimm2010\"] = 1,\n [\"CITEREFDowker2019\"] = 1,\n [\"CITEREFDreeben-Irimia2010\"] = 1,\n [\"CITEREFDrosg2007\"] = 1,\n [\"CITEREFDuffy2018\"] = 1,\n [\"CITEREFDuverney2010\"] = 1,\n [\"CITEREFEbbyHulbertBroadhead2020\"] = 1,\n [\"CITEREFEmersonBabtie2014\"] = 1,\n [\"CITEREFErikssonEstepJohnson2013\"] = 1,\n [\"CITEREFFarmer2023\"] = 1,\n [\"CITEREFFerreiros2013\"] = 1,\n [\"CITEREFGallistelGelman2005\"] = 1,\n [\"CITEREFGeary2006\"] = 1,\n [\"CITEREFGellertHellwichKästnerKüstner2012\"] = 1,\n [\"CITEREFGerardiGoetteMeier2013\"] = 1,\n [\"CITEREFGoodstein2014\"] = 1,\n [\"CITEREFGriceKempMortonGrace2023\"] = 1,\n [\"CITEREFGriffin1935\"] = 1,\n [\"CITEREFGrigorieva2018\"] = 1,\n [\"CITEREFGupta2019\"] = 1,\n [\"CITEREFHC_staff2022\"] = 1,\n [\"CITEREFHC_staff2022a\"] = 1,\n [\"CITEREFHC_staff2022b\"] = 1,\n [\"CITEREFHafstrom2013\"] = 1,\n [\"CITEREFHamiltonLandin2018\"] = 1,\n [\"CITEREFHart2011\"] = 1,\n [\"CITEREFHaylockCockburn2008\"] = 1,\n [\"CITEREFHigham2002\"] = 1,\n [\"CITEREFHindry2011\"] = 1,\n [\"CITEREFHodgkin2013\"] = 1,\n [\"CITEREFHoffmanFrankel2018\"] = 1,\n [\"CITEREFHofweber2016\"] = 1,\n [\"CITEREFHorsten2023\"] = 1,\n [\"CITEREFHuaFeng2020\"] = 1,\n [\"CITEREFHusserlWillard2012\"] = 1,\n [\"CITEREFITL_Education_Solutions_Limited2011\"] = 1,\n [\"CITEREFIgarashiAltmanFunadaKamiyama2014\"] = 1,\n [\"CITEREFInternational_Organization_for_Standardization2019\"] = 1,\n [\"CITEREFJackson2008\"] = 1,\n [\"CITEREFJena2021\"] = 1,\n [\"CITEREFKaiserGranade2021\"] = 1,\n [\"CITEREFKaratsuba2014\"] = 1,\n [\"CITEREFKaratsuba2020\"] = 1,\n [\"CITEREFKay2021\"] = 1,\n [\"CITEREFKhanGraham2018\"] = 1,\n [\"CITEREFKhattar2010\"] = 1,\n [\"CITEREFKhouryLamothe2016\"] = 1,\n [\"CITEREFKlaf2011\"] = 1,\n [\"CITEREFKlein2013\"] = 1,\n [\"CITEREFKlein2013a\"] = 1,\n [\"CITEREFKleinMoellerDresselDomahs2010\"] = 1,\n [\"CITEREFKleiner2012\"] = 1,\n [\"CITEREFKlose2014\"] = 1,\n [\"CITEREFKnoblochKomatsuLiu2013\"] = 1,\n [\"CITEREFKoepf2021\"] = 1,\n [\"CITEREFKoetsier2018\"] = 1,\n [\"CITEREFKoren2018\"] = 1,\n [\"CITEREFKrennLorünser2023\"] = 1,\n [\"CITEREFKubilyus2018\"] = 1,\n [\"CITEREFKudryavtsev2020\"] = 1,\n [\"CITEREFKupferman2015\"] = 1,\n [\"CITEREFKörner2019\"] = 1,\n [\"CITEREFKřížekSomerŠolcová2021\"] = 1,\n [\"CITEREFLang2002\"] = 1,\n [\"CITEREFLang2015\"] = 1,\n [\"CITEREFLange2010\"] = 1,\n [\"CITEREFLaskiJor’danDaoustMurray2015\"] = 1,\n [\"CITEREFLernerLerner2008\"] = 1,\n [\"CITEREFLiSchoenfeld2019\"] = 1,\n [\"CITEREFLiebler2018\"] = 1,\n [\"CITEREFLockhart2017\"] = 1,\n [\"CITEREFLozano-Robledo2019\"] = 1,\n [\"CITEREFLudererNollauVetters2013\"] = 1,\n [\"CITEREFLustick1997\"] = 1,\n [\"CITEREFLützen2023\"] = 1,\n [\"CITEREFMW_staff2023\"] = 1,\n [\"CITEREFMa2020\"] = 1,\n [\"CITEREFMaddenAubrey2017\"] = 1,\n [\"CITEREFMahajan2010\"] = 1,\n [\"CITEREFMarcusMcEvoy2016\"] = 1,\n [\"CITEREFMazumderEbong2023\"] = 1,\n [\"CITEREFMazzolaMilmeisterWeissmann2004\"] = 1,\n [\"CITEREFMeyer2023\"] = 1,\n [\"CITEREFMonahan2012\"] = 1,\n [\"CITEREFMoncayo2018\"] = 1,\n [\"CITEREFMooneyBriggsHansenMcCullouch2014\"] = 1,\n [\"CITEREFMooreKearfottCloud2009\"] = 1,\n [\"CITEREFMullerBrisebarreDinechinJeannerod2009\"] = 1,\n [\"CITEREFMullerBrunieDinechinJeannerod2018\"] = 1,\n [\"CITEREFMusserPetersonBurger2013\"] = 1,\n [\"CITEREFNCTM_Staff\"] = 1,\n [\"CITEREFNagel2002\"] = 1,\n [\"CITEREFNagelNewman2008\"] = 1,\n [\"CITEREFNakovKolev2013\"] = 1,\n [\"CITEREFNelson2019\"] = 1,\n [\"CITEREFNullLobur2006\"] = 1,\n [\"CITEREFNurnberger-Haag2017\"] = 1,\n [\"CITEREFO\u0026#039;Leary2015\"] = 1,\n [\"CITEREFO\u0026#039;Regan2012\"] = 1,\n [\"CITEREFOakes2020\"] = 1,\n [\"CITEREFOdomBarbarinWasik2009\"] = 1,\n [\"CITEREFOliver2005\"] = 1,\n [\"CITEREFOmondi2020\"] = 1,\n [\"CITEREFOngleyCarey2013\"] = 1,\n [\"CITEREFOre1948\"] = 1,\n [\"CITEREFOrr1995\"] = 1,\n [\"CITEREFOtis2024\"] = 1,\n [\"CITEREFPaarPelzl2009\"] = 1,\n [\"CITEREFPage2003\"] = 1,\n [\"CITEREFPayne2017\"] = 1,\n [\"CITEREFPeirce2015\"] = 1,\n [\"CITEREFPharrJakobHumphreys2023\"] = 1,\n [\"CITEREFPomerance2010\"] = 1,\n [\"CITEREFPomeranceSárközy1995\"] = 1,\n [\"CITEREFPonticorvoSchmbriMiglino2019\"] = 1,\n [\"CITEREFPrata2002\"] = 1,\n [\"CITEREFQuinteroRosario2016\"] = 1,\n [\"CITEREFRajan2022\"] = 1,\n [\"CITEREFReilly2009\"] = 1,\n [\"CITEREFReitano2010\"] = 1,\n [\"CITEREFResnickFord2012\"] = 1,\n [\"CITEREFReynolds2008\"] = 1,\n [\"CITEREFRiesel2012\"] = 1,\n [\"CITEREFRisingMatthewsSchoaffMatthew2021\"] = 1,\n [\"CITEREFRobbins2006\"] = 1,\n [\"CITEREFRoddaLittle2015\"] = 1,\n [\"CITEREFRoedeForestJamshidi2018\"] = 1,\n [\"CITEREFRomanowski2008\"] = 1,\n [\"CITEREFRooney2021\"] = 1,\n [\"CITEREFRossi2011\"] = 1,\n [\"CITEREFRuthven2012\"] = 1,\n [\"CITEREFSallySally_(Jr.)2012\"] = 1,\n [\"CITEREFSeamanRosslerBurgin2023\"] = 1,\n [\"CITEREFShiva2018\"] = 1,\n [\"CITEREFSierpinskaLerman1996\"] = 1,\n [\"CITEREFSmith1958\"] = 1,\n [\"CITEREFSmyth1864\"] = 1,\n [\"CITEREFSophian2017\"] = 1,\n [\"CITEREFSperlingStuart1981\"] = 1,\n [\"CITEREFStakhov2020\"] = 1,\n [\"CITEREFSternbergBen-Zeev2012\"] = 1,\n [\"CITEREFStevensonWaite2011\"] = 1,\n [\"CITEREFStewart2022\"] = 1,\n [\"CITEREFStrathern2012\"] = 1,\n [\"CITEREFSwanson2021\"] = 1,\n [\"CITEREFSwartzlander2017\"] = 1,\n [\"CITEREFTarasov2008\"] = 1,\n [\"CITEREFTaylor2012\"] = 1,\n [\"CITEREFThiamRochon2019\"] = 1,\n [\"CITEREFTiles2009\"] = 1,\n [\"CITEREFUspenskiiSemenov2001\"] = 1,\n [\"CITEREFVaccaroPepiciello2022\"] = 1,\n [\"CITEREFVerschaffelTorbeynsDe_Smedt2011\"] = 1,\n [\"CITEREFVictoria_Department_of_Education_Staff2023\"] = 1,\n [\"CITEREFVinogradov2019\"] = 1,\n [\"CITEREFVullo2020\"] = 1,\n [\"CITEREFWaite2013\"] = 1,\n [\"CITEREFWallis2011\"] = 1,\n [\"CITEREFWallis2013\"] = 1,\n [\"CITEREFWang1997\"] = 1,\n [\"CITEREFWard2012\"] = 1,\n [\"CITEREFWedell2015\"] = 1,\n [\"CITEREFWeil2009\"] = 1,\n [\"CITEREFWeir2022\"] = 1,\n [\"CITEREFWheater2015\"] = 1,\n [\"CITEREFWilson2020\"] = 1,\n [\"CITEREFWrightEllemor-CollinsTabor2011\"] = 1,\n [\"CITEREFXuZhang2022\"] = 1,\n [\"CITEREFYadin2016\"] = 1,\n [\"CITEREFYan2002\"] = 1,\n [\"CITEREFYan2013a\"] = 1,\n [\"CITEREFYoung2010\"] = 1,\n [\"CITEREFYoung2021\"] = 1,\n [\"CITEREFZhang2012\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 1,\n [\"Areas of mathematics\"] = 1,\n [\"Authority control\"] = 1,\n [\"Cite book\"] = 190,\n [\"Cite journal\"] = 9,\n [\"Cite web\"] = 19,\n [\"Clear\"] = 1,\n [\"Efn\"] = 10,\n [\"For-multi\"] = 1,\n [\"Gloss\"] = 2,\n [\"Good article\"] = 1,\n [\"Harvnb\"] = 463,\n [\"Main\"] = 7,\n [\"Multiple image\"] = 6,\n [\"Multiref\"] = 164,\n [\"Nobr\"] = 5,\n [\"Notelist\"] = 1,\n [\"Overline\"] = 1,\n [\"Pi\"] = 6,\n [\"Portal\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn\"] = 1,\n [\"Short description\"] = 1,\n [\"Slink\"] = 1,\n [\"Subject bar\"] = 1,\n [\"Tmath\"] = 2,\n [\"Wikt-lang\"] = 4,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n","limitreport-profile":[["?","340","21.5"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","260","16.5"],["recursiveClone \u003CmwInit.lua:45\u003E","140","8.9"],["type","100","6.3"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::find","100","6.3"],["dataWrapper \u003Cmw.lua:672\u003E","80","5.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::anchorEncode","60","3.8"],["\u003Cmw.lua:694\u003E","60","3.8"],["pairs","40","2.5"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","40","2.5"],["[others]","360","22.8"]]},"cachereport":{"origin":"mw-web.codfw.next-6567bdfc66-h6wzd","timestamp":"20250302202045","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Arithmetic","url":"https:\/\/en.wikipedia.org\/wiki\/Arithmetic","sameAs":"http:\/\/www.wikidata.org\/entity\/Q11205","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q11205","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-01-05T22:00:55Z","dateModified":"2025-01-28T09:16:53Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/4\/45\/Arithmetic_operations.svg","headline":"elementary branch of mathematics"}</script> </body> </html>