CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Number - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-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":"769fb758-703f-4902-8a35-626dbbe5ced7","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Number","wgTitle":"Number","wgCurRevisionId":1256779505,"wgRevisionId":1256779505,"wgArticleId":21690,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: location missing publisher","Webarchive template wayback links","Articles with short description","Short description is different from Wikidata","Wikipedia indefinitely semi-protected pages","Wikipedia indefinitely move-protected pages","Use dmy dates from December 2022","All articles lacking reliable references","Articles lacking reliable references from January 2017","Articles needing additional references from November 2022","All articles needing additional references", "Articles containing Sanskrit-language text","Articles lacking reliable references from September 2020","Articles containing Latin-language text","All articles with unsourced statements","Articles with unsourced statements from September 2020","Articles containing German-language text","Wikipedia articles needing clarification from September 2020","Pages with missing ISBNs","Articles prone to spam from July 2022","Commons link is on Wikidata","Numbers","Group theory","Abstraction","Mathematical objects"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Number","wgRelevantArticleId":21690,"wgIsProbablyEditable":false,"wgRelevantPageIsProbablyEditable":false,"wgRestrictionEdit":["autoconfirmed"],"wgRestrictionMove":["sysop"],"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":70000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q11563","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","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","wikibase.sidebar.tracking"];</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.4"> <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/a/a0/NumberSetinC.svg/1200px-NumberSetinC.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="940"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/800px-NumberSetinC.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="627"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/640px-NumberSetinC.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="501"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Number - 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/Number"> <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/Number"> <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 page-Number rootpage-Number 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" > <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> </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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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=Number" 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=Number" 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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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=Number" 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=Number" 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-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">1</span> <span>History</span> </div> </a> <button aria-controls="toc-History-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 History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-First_use_of_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_use_of_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>First use of numbers</span> </div> </a> <ul id="toc-First_use_of_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Numerals</span> </div> </a> <ul id="toc-Numerals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zero"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Zero</span> </div> </a> <ul id="toc-Zero-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negative_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Negative numbers</span> </div> </a> <ul id="toc-Negative_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Rational numbers</span> </div> </a> <ul id="toc-Rational_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irrational_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irrational_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Irrational numbers</span> </div> </a> <ul id="toc-Irrational_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transcendental_numbers_and_reals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transcendental_numbers_and_reals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Transcendental numbers and reals</span> </div> </a> <ul id="toc-Transcendental_numbers_and_reals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinity_and_infinitesimals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinity_and_infinitesimals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Infinity and infinitesimals</span> </div> </a> <ul id="toc-Infinity_and_infinitesimals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.10</span> <span>Prime numbers</span> </div> </a> <ul id="toc-Prime_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_classification" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Main_classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Main classification</span> </div> </a> <button aria-controls="toc-Main_classification-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 Main classification subsection</span> </button> <ul id="toc-Main_classification-sublist" class="vector-toc-list"> <li id="toc-Natural_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Natural numbers</span> </div> </a> <ul id="toc-Natural_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Integers</span> </div> </a> <ul id="toc-Integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_numbers_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_numbers_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Rational numbers</span> </div> </a> <ul id="toc-Rational_numbers_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Real numbers</span> </div> </a> <ul id="toc-Real_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_numbers_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subclasses_of_the_integers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subclasses_of_the_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Subclasses of the integers</span> </div> </a> <button aria-controls="toc-Subclasses_of_the_integers-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 Subclasses of the integers subsection</span> </button> <ul id="toc-Subclasses_of_the_integers-sublist" class="vector-toc-list"> <li id="toc-Even_and_odd_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Even_and_odd_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Even and odd numbers</span> </div> </a> <ul id="toc-Even_and_odd_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_numbers_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_numbers_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Prime numbers</span> </div> </a> <ul id="toc-Prime_numbers_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_classes_of_integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_classes_of_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Other classes of integers</span> </div> </a> <ul id="toc-Other_classes_of_integers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subclasses_of_the_complex_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subclasses_of_the_complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Subclasses of the complex numbers</span> </div> </a> <button aria-controls="toc-Subclasses_of_the_complex_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 Subclasses of the complex numbers subsection</span> </button> <ul id="toc-Subclasses_of_the_complex_numbers-sublist" class="vector-toc-list"> <li id="toc-Algebraic,_irrational_and_transcendental_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic,_irrational_and_transcendental_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Algebraic, irrational and transcendental numbers</span> </div> </a> <ul id="toc-Algebraic,_irrational_and_transcendental_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Periods_and_exponential_periods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Periods_and_exponential_periods"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Periods and exponential periods</span> </div> </a> <ul id="toc-Periods_and_exponential_periods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructible_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constructible_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Constructible numbers</span> </div> </a> <ul id="toc-Constructible_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computable_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computable_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Computable numbers</span> </div> </a> <ul id="toc-Computable_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions_of_the_concept" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions_of_the_concept"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extensions of the concept</span> </div> </a> <button aria-controls="toc-Extensions_of_the_concept-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 Extensions of the concept subsection</span> </button> <ul id="toc-Extensions_of_the_concept-sublist" class="vector-toc-list"> <li id="toc-p-adic_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#p-adic_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span><i>p</i>-adic numbers</span> </div> </a> <ul id="toc-p-adic_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypercomplex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypercomplex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Hypercomplex numbers</span> </div> </a> <ul id="toc-Hypercomplex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transfinite_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transfinite_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Transfinite numbers</span> </div> </a> <ul id="toc-Transfinite_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonstandard_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonstandard_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Nonstandard numbers</span> </div> </a> <ul id="toc-Nonstandard_numbers-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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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">9</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" > <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">Number</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 189 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-189" 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">189 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/Getal" title="Getal – Afrikaans" lang="af" hreflang="af" data-title="Getal" 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/Zahl" title="Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Zahl" 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%89%81%E1%8C%A5%E1%88%AD" title="ቁጥር – Amharic" lang="am" hreflang="am" data-title="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%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – Angika" lang="anp" hreflang="anp" data-title="संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/R%C4%ABm" title="Rīm – Old English" lang="ang" hreflang="ang" data-title="Rīm" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF" title="عدد – Arabic" lang="ar" hreflang="ar" data-title="عدد" 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/Numero" title="Numero – Aragonese" lang="an" hreflang="an" data-title="Numero" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%A1%DC%A2%DC%9D%DC%A2%DC%90" title="ܡܢܝܢܐ – Aramaic" lang="arc" hreflang="arc" data-title="ܡܢܝܢܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B9%D5%AB%D6%82" title="Թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/Numiru" title="Numiru – Aromanian" lang="rup" hreflang="rup" data-title="Numiru" data-language-autonym="Armãneashti" data-language-local-name="Aromanian" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-frp mw-list-item"><a href="https://frp.wikipedia.org/wiki/Nombro" title="Nombro – Arpitan" lang="frp" hreflang="frp" data-title="Nombro" data-language-autonym="Arpetan" data-language-local-name="Arpitan" class="interlanguage-link-target"><span>Arpetan</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="সংখ্যা – Assamese" lang="as" hreflang="as" data-title="সংখ্যা" 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/N%C3%BAmberu" title="Númberu – Asturian" lang="ast" hreflang="ast" data-title="Númberu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-atj mw-list-item"><a href="https://atj.wikipedia.org/wiki/Akitasowin" title="Akitasowin – Atikamekw" lang="atj" hreflang="atj" data-title="Akitasowin" data-language-autonym="Atikamekw" data-language-local-name="Atikamekw" class="interlanguage-link-target"><span>Atikamekw</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapy" title="Papapy – Guarani" lang="gn" hreflang="gn" data-title="Papapy" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99d" title="Ədəd – Azerbaijani" lang="az" hreflang="az" data-title="Ədəd" 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%B9%D8%AF%D8%AF" title="عدد – South Azerbaijani" lang="azb" hreflang="azb" data-title="عدد" 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%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Wilangan" title="Wilangan – Banjar" lang="bjn" hreflang="bjn" data-title="Wilangan" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-ba%CC%8Dk" title="Sò͘-ba̍k – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-ba̍k" 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 mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D0%B0%D0%BD" title="Һан – Bashkir" lang="ba" hreflang="ba" data-title="Һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA" title="Лік – Belarusian" lang="be" hreflang="be" data-title="Лік" 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%9B%D1%96%D0%BA" title="Лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лік" 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%A8%E0%A4%82%E0%A4%AC%E0%A4%B0" title="नंबर – Bhojpuri" lang="bh" hreflang="bh" data-title="नंबर" 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/Bilang" title="Bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Bilang" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – Bulgarian" lang="bg" hreflang="bg" data-title="Число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Zoih" title="Zoih – Bavarian" lang="bar" hreflang="bar" data-title="Zoih" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%82%E0%BE%B2%E0%BD%84%E0%BD%A6%E0%BC%8B%E0%BD%80%E0%BC%8D" title="གྲངས་ཀ། – Tibetan" lang="bo" hreflang="bo" data-title="གྲངས་ཀ།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Broj" title="Broj – Bosnian" lang="bs" hreflang="bs" data-title="Broj" 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/Niver" title="Niver – Breton" lang="br" hreflang="br" data-title="Niver" 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%A2%D0%BE%D0%BE" title="Тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Тоо" 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 badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ca.wikipedia.org/wiki/Nombre" title="Nombre – Catalan" lang="ca" hreflang="ca" data-title="Nombre" 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%A5%D0%B8%D1%81%D0%B5%D0%BF" title="Хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8C%C3%ADslo" title="Číslo – Czech" lang="cs" hreflang="cs" data-title="Číslo" 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/Numero" title="Numero – Chavacano" lang="cbk" hreflang="cbk" data-title="Numero" 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/Nhamba" title="Nhamba – Shona" lang="sn" hreflang="sn" data-title="Nhamba" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-tum mw-list-item"><a href="https://tum.wikipedia.org/wiki/Chipendero" title="Chipendero – Tumbuka" lang="tum" hreflang="tum" data-title="Chipendero" data-language-autonym="ChiTumbuka" data-language-local-name="Tumbuka" class="interlanguage-link-target"><span>ChiTumbuka</span></a></li><li class="interlanguage-link interwiki-cho mw-list-item"><a href="https://cho.wikipedia.org/wiki/Hohltina" title="Hohltina – Choctaw" lang="cho" hreflang="cho" data-title="Hohltina" data-language-autonym="Chahta anumpa" data-language-local-name="Choctaw" class="interlanguage-link-target"><span>Chahta anumpa</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Numeru" title="Numeru – Corsican" lang="co" hreflang="co" data-title="Numeru" 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/Rhif" title="Rhif – Welsh" lang="cy" hreflang="cy" data-title="Rhif" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-dag mw-list-item"><a href="https://dag.wikipedia.org/wiki/Kalinli" title="Kalinli – Dagbani" lang="dag" hreflang="dag" data-title="Kalinli" data-language-autonym="Dagbanli" data-language-local-name="Dagbani" class="interlanguage-link-target"><span>Dagbanli</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tal" title="Tal – Danish" lang="da" hreflang="da" data-title="Tal" 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/Zahl" title="Zahl – German" lang="de" hreflang="de" data-title="Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dty mw-list-item"><a href="https://dty.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95%E0%A4%BE" title="अंका – Doteli" lang="dty" hreflang="dty" data-title="अंका" data-language-autonym="डोटेली" data-language-local-name="Doteli" class="interlanguage-link-target"><span>डोटेली</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arv" title="Arv – Estonian" lang="et" hreflang="et" data-title="Arv" 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%CF%8C%CF%82" title="Αριθμός – Greek" lang="el" hreflang="el" data-title="Αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer" title="Nómmer – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero" title="Número – Spanish" lang="es" hreflang="es" data-title="Número" 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/Nombro" title="Nombro – Esperanto" lang="eo" hreflang="eo" data-title="Nombro" 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/N%C3%BAmiru" title="Númiru – Extremaduran" lang="ext" hreflang="ext" data-title="Númiru" 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/Zenbaki" title="Zenbaki – Basque" lang="eu" hreflang="eu" data-title="Zenbaki" 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%B9%D8%AF%D8%AF" title="عدد – Persian" lang="fa" hreflang="fa" data-title="عدد" 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/Ginti" title="Ginti – Fiji Hindi" lang="hif" hreflang="hif" data-title="Ginti" 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-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Tal" title="Tal – Faroese" lang="fo" hreflang="fo" data-title="Tal" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre" title="Nombre – French" lang="fr" hreflang="fr" data-title="Nombre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Getal" title="Getal – Western Frisian" lang="fy" hreflang="fy" data-title="Getal" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ff mw-list-item"><a href="https://ff.wikipedia.org/wiki/Limle" title="Limle – Fula" lang="ff" hreflang="ff" data-title="Limle" data-language-autonym="Fulfulde" data-language-local-name="Fula" class="interlanguage-link-target"><span>Fulfulde</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir" title="Uimhir – Irish" lang="ga" hreflang="ga" data-title="Uimhir" 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%80ireamh" title="Àireamh – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Àireamh" 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/N%C3%BAmero" title="Número – Galician" lang="gl" hreflang="gl" data-title="Número" 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/%E6%95%B8" title="數 – Gan" lang="gan" hreflang="gan" data-title="數" 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-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D0%B3" title="Тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Тойг" 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%88%98_(%EC%88%98%ED%95%99)" title="수 (수학) – Korean" lang="ko" hreflang="ko" data-title="수 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Lamba" title="Lamba – Hausa" lang="ha" hreflang="ha" data-title="Lamba" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/N%C4%81_helu" title="Nā helu – Hawaiian" lang="haw" hreflang="haw" data-title="Nā helu" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target"><span>Hawaiʻi</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%AB%D5%BE" title="Թիվ – Armenian" lang="hy" hreflang="hy" data-title="Թիվ" 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%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – Hindi" lang="hi" hreflang="hi" data-title="संख्या" 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/Broj" title="Broj – Croatian" lang="hr" hreflang="hr" data-title="Broj" 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/Nombro" title="Nombro – Ido" lang="io" hreflang="io" data-title="Nombro" 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/Numero" title="Numero – Iloko" lang="ilo" hreflang="ilo" data-title="Numero" 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/Bilangan" title="Bilangan – Indonesian" lang="id" hreflang="id" data-title="Bilangan" 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/Numero" title="Numero – Interlingua" lang="ia" hreflang="ia" data-title="Numero" 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%9D%D1%8B%D0%BC%C3%A6%D1%86" title="Нымæц – Ossetic" lang="os" hreflang="os" data-title="Нымæц" 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/INANI" title="INANI – Xhosa" lang="xh" hreflang="xh" data-title="INANI" 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/Inombolo" title="Inombolo – Zulu" lang="zu" hreflang="zu" data-title="Inombolo" 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/Tala_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Tala (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Tala (stærðfræði)" 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/Numero" title="Numero – Italian" lang="it" hreflang="it" data-title="Numero" 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%9E%D7%A1%D7%A4%D7%A8" title="מספר – Hebrew" lang="he" hreflang="he" data-title="מספר" 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/Wilangan_(mat%C3%A9matika)" title="Wilangan (matématika) – Javanese" lang="jv" hreflang="jv" data-title="Wilangan (matématika)" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/%C3%91%CA%8A%CA%8A_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Ñʊʊ (tʊʊzʊʊ) – Kabiye" lang="kbp" hreflang="kbp" data-title="Ñʊʊ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಸಂಖ್ಯೆ – Kannada" lang="kn" hreflang="kn" data-title="ಸಂಖ್ಯೆ" 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%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="რიცხვი" 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%A1%D0%B0%D0%BD" title="Сан – Kazakh" lang="kk" hreflang="kk" data-title="Сан" 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/Niver" title="Niver – Cornish" lang="kw" hreflang="kw" data-title="Niver" 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/Namba" title="Namba – Swahili" lang="sw" hreflang="sw" data-title="Namba" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Nonm" title="Nonm – Haitian Creole" lang="ht" hreflang="ht" data-title="Nonm" 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/Nomm" title="Nomm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nomm" 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-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar" title="Hejmar – Kurdish" lang="ku" hreflang="ku" data-title="Hejmar" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lbe mw-list-item"><a href="https://lbe.wikipedia.org/wiki/%D0%90%D1%8C%D0%B4%D0%B0%D0%B4" title="Аьдад – Lak" lang="lbe" hreflang="lbe" data-title="Аьдад" data-language-autonym="Лакку" data-language-local-name="Lak" class="interlanguage-link-target"><span>Лакку</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99" title="ຈຳນວນ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus" title="Numerus – Latin" lang="la" hreflang="la" data-title="Numerus" 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/Skaitlis" title="Skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Zuel" title="Zuel – Luxembourgish" lang="lb" hreflang="lb" data-title="Zuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skai%C4%8Dius" title="Skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Getal" title="Getal – Limburgish" lang="li" hreflang="li" data-title="Getal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero" title="Numero – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero" 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/namcu" title="namcu – Lojban" lang="jbo" hreflang="jbo" data-title="namcu" 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-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Ennamba" title="Ennamba – Ganda" lang="lg" hreflang="lg" data-title="Ennamba" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer" title="Numer – Lombard" lang="lmo" hreflang="lmo" data-title="Numer" 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%A1m" title="Szám – Hungarian" lang="hu" hreflang="hu" data-title="Szám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%95" title="अंक – Maithili" lang="mai" hreflang="mai" data-title="अंक" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target"><span>मैथिली</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%98" title="Број – Macedonian" lang="mk" hreflang="mk" data-title="Број" 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/Isa" title="Isa – Malagasy" lang="mg" hreflang="mg" data-title="Isa" 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%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="സംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="സംഖ്യ" 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%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संख्या – Marathi" lang="mr" hreflang="mr" data-title="संख्या" 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%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%A3" title="რიცხუ – Mingrelian" lang="xmf" hreflang="xmf" data-title="რიცხუ" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%A7%D8%B4%D9%85%D8%A7%D8%B1%DA%A9" title="اشمارک – Mazanderani" lang="mzn" hreflang="mzn" data-title="اشمارک" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor" title="Nombor – Malay" lang="ms" hreflang="ms" data-title="Nombor" 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/N%C3%BAmaro" title="Númaro – Mirandese" lang="mwl" hreflang="mwl" data-title="Númaro" 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%A2%D0%BE%D0%BE" title="Тоо – Mongolian" lang="mn" hreflang="mn" data-title="Тоо" 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%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="ကိန်း – Burmese" lang="my" hreflang="my" data-title="ကိန်း" 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/Tlapohualli" title="Tlapohualli – Nahuatl" lang="nah" hreflang="nah" data-title="Tlapohualli" 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/Naba" title="Naba – Fijian" lang="fj" hreflang="fj" data-title="Naba" 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/Getal_(wiskunde)" title="Getal (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Getal (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-cr mw-list-item"><a href="https://cr.wikipedia.org/wiki/%E1%90%8A%E1%91%AD%E1%90%A6%E1%91%96%E1%93%B1%E1%90%A3" title="ᐊᑭᐦᑖᓱᐣ – Cree" lang="cr" hreflang="cr" data-title="ᐊᑭᐦᑖᓱᐣ" data-language-autonym="Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ" data-language-local-name="Cree" class="interlanguage-link-target"><span>Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ</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" title="अङ्क – Nepali" lang="ne" hreflang="ne" data-title="अङ्क" 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%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83" title="ल्याः – Newari" lang="new" hreflang="new" data-title="ल्याः" 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/%E6%95%B0" title="数 – Japanese" lang="ja" hreflang="ja" data-title="数" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Taal" title="Taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Taal" 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/Tall" title="Tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tall" 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/Tal" title="Tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tal" 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-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Neunm%C3%A9tho" title="Neunmétho – Norman" lang="nrf" hreflang="nrf" data-title="Neunmétho" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target"><span>Nouormand</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Nombre" title="Nombre – Novial" lang="nov" hreflang="nov" data-title="Nombre" 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/Nombre" title="Nombre – Occitan" lang="oc" hreflang="oc" data-title="Nombre" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%A8%D0%BE%D1%82%D0%BF%D0%B0%D0%BB" title="Шотпал – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Шотпал" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Son" title="Son – Uzbek" lang="uz" hreflang="uz" data-title="Son" 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" title="ਅੰਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਅੰਕ" 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/%D9%86%D9%85%D8%A8%D8%B1" title="نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF" title="عدد – Pashto" lang="ps" hreflang="ps" data-title="عدد" 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/Nomba" title="Nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nomba" 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-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Tahl" title="Tahl – Low German" lang="nds" hreflang="nds" data-title="Tahl" 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/Liczba" title="Liczba – Polish" lang="pl" hreflang="pl" data-title="Liczba" 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/N%C3%BAmero" title="Número – Portuguese" lang="pt" hreflang="pt" data-title="Número" 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/San" title="San – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="San" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r" title="Număr – Romanian" lang="ro" hreflang="ro" data-title="Număr" 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/Yupay" title="Yupay – Quechua" lang="qu" hreflang="qu" data-title="Yupay" 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%A7%D1%96%D1%81%D0%BB%D0%BE" title="Чісло – Rusyn" lang="rue" hreflang="rue" data-title="Чісло" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – Russian" lang="ru" hreflang="ru" data-title="Число" 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%85%D1%81%D0%B0%D0%B0%D0%BD" title="Ахсаан – Yakut" lang="sah" hreflang="sah" data-title="Ахсаан" 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%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83" title="संख्याः – Sanskrit" lang="sa" hreflang="sa" data-title="संख्याः" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sg mw-list-item"><a href="https://sg.wikipedia.org/wiki/N%C3%B6m%C3%B6r%C3%B6" title="Nömörö – Sango" lang="sg" hreflang="sg" data-title="Nömörö" data-language-autonym="Sängö" data-language-local-name="Sango" class="interlanguage-link-target"><span>Sängö</span></a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Nomoro" title="Nomoro – Northern Sotho" lang="nso" hreflang="nso" data-title="Nomoro" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Northern Sotho" class="interlanguage-link-target"><span>Sesotho sa Leboa</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri" title="Numri – Albanian" lang="sq" hreflang="sq" data-title="Numri" 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/N%C3%B9mmuru" title="Nùmmuru – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru" 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/Number" title="Number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Number" 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%D9%86%DA%AF" title="انگ – Sindhi" lang="sd" hreflang="sd" data-title="انگ" 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/%C4%8C%C3%ADslo_(matematika)" title="Číslo (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Číslo (matematika)" 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/%C5%A0tevilo" title="Število – Slovenian" lang="sl" hreflang="sl" data-title="Število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BC%D1%A7" title="Чисмѧ – Church Slavic" lang="cu" hreflang="cu" data-title="Чисмѧ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/N%C5%AFmera" title="Nůmera – Silesian" lang="szl" hreflang="szl" data-title="Nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_(cillanaad)" title="Thiin (cillanaad) – Somali" lang="so" hreflang="so" data-title="Thiin (cillanaad)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95" title="ژمارە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارە" 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%91%D1%80%D0%BE%D1%98" title="Број – Serbian" lang="sr" hreflang="sr" data-title="Број" 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/Broj" title="Broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Broj" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Wilangan" title="Wilangan – Sundanese" lang="su" hreflang="su" data-title="Wilangan" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luku" title="Luku – Finnish" lang="fi" hreflang="fi" data-title="Luku" 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/Tal" title="Tal – Swedish" lang="sv" hreflang="sv" data-title="Tal" 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/Bilang" title="Bilang – Tagalog" lang="tl" hreflang="tl" data-title="Bilang" 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" title="எண் – Tamil" lang="ta" hreflang="ta" data-title="எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Am%E1%B8%8Dan" title="Amḍan – Tachelhit" lang="shi" hreflang="shi" data-title="Amḍan" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Am%E1%B8%8Dan" title="Amḍan – Kabyle" lang="kab" hreflang="kab" data-title="Amḍan" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-roa-tara mw-list-item"><a href="https://roa-tara.wikipedia.org/wiki/Numere" title="Numere – Tarantino" lang="nap-x-tara" hreflang="nap-x-tara" data-title="Numere" data-language-autonym="Tarandíne" data-language-local-name="Tarantino" class="interlanguage-link-target"><span>Tarandíne</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BD" title="Сан – Tatar" lang="tt" hreflang="tt" data-title="Сан" 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-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="సంఖ్య – Telugu" lang="te" hreflang="te" data-title="సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99" title="จำนวน – Thai" lang="th" hreflang="th" data-title="จำนวน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-ti mw-list-item"><a href="https://ti.wikipedia.org/wiki/%E1%89%81%E1%8C%BD%E1%88%AA" title="ቁጽሪ – Tigrinya" lang="ti" hreflang="ti" data-title="ቁጽሪ" data-language-autonym="ትግርኛ" data-language-local-name="Tigrinya" class="interlanguage-link-target"><span>ትግርኛ</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4" title="Адад – Tajik" lang="tg" hreflang="tg" data-title="Адад" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tcy mw-list-item"><a href="https://tcy.wikipedia.org/wiki/%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಸಂಖ್ಯೆ – Tulu" lang="tcy" hreflang="tcy" data-title="ಸಂಖ್ಯೆ" data-language-autonym="ತುಳು" data-language-local-name="Tulu" class="interlanguage-link-target"><span>ತುಳು</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Say%C4%B1" title="Sayı – Turkish" lang="tr" hreflang="tr" data-title="Sayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/San" title="San – Turkmen" lang="tk" hreflang="tk" data-title="San" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%9B%D1%8B%D0%B4" title="Лыд – Udmurt" lang="udm" hreflang="udm" data-title="Лыд" data-language-autonym="Удмурт" data-language-local-name="Udmurt" class="interlanguage-link-target"><span>Удмурт</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число – Ukrainian" lang="uk" hreflang="uk" data-title="Число" 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%B9%D8%AF%D8%AF" title="عدد – Urdu" lang="ur" hreflang="ur" data-title="عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/N%C3%B9maro" title="Nùmaro – Venetian" lang="vec" hreflang="vec" data-title="Nùmaro" 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/Lugu" title="Lugu – Veps" lang="vep" hreflang="vep" data-title="Lugu" 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" title="Số – Vietnamese" lang="vi" hreflang="vi" data-title="Số" 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-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Arv" title="Arv – Võro" lang="vro" hreflang="vro" data-title="Arv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-guc mw-list-item"><a href="https://guc.wikipedia.org/wiki/Nuumerokana" title="Nuumerokana – Wayuu" lang="guc" hreflang="guc" data-title="Nuumerokana" data-language-autonym="Wayuunaiki" data-language-local-name="Wayuu" class="interlanguage-link-target"><span>Wayuunaiki</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B8" title="數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="數" 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/Ihap" title="Ihap – Waray" lang="war" hreflang="war" data-title="Ihap" 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/%E6%95%B0" title="数 – Wu" lang="wuu" hreflang="wuu" data-title="数" 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/Nomboro" title="Nomboro – Tsonga" lang="ts" hreflang="ts" data-title="Nomboro" 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%A6%D7%90%D7%9C" title="צאל – Yiddish" lang="yi" hreflang="yi" data-title="צאל" 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/N%E1%BB%8D%CC%81mb%C3%A0" title="Nọ́mbà – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà" 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/%E6%95%B8" title="數 – Cantonese" lang="yue" hreflang="yue" data-title="數" 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-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Skaitlios" title="Skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Skaitlios" 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/%E6%95%B0" title="数 – Chinese" lang="zh" hreflang="zh" data-title="数" 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/Lumur" title="Lumur – Iban" lang="iba" hreflang="iba" data-title="Lumur" 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-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%8E%E2%B4%B9%E2%B4%B0%E2%B5%8F" title="ⴰⵎⴹⴰⵏ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⴰⵎⴹⴰⵏ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</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/Q11563#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/Number" 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:Number" 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/Number"><span>Read</span></a></li><li id="ca-viewsource" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Number&action=edit" title="This page is protected.
You can view its source [e]" accesskey="e"><span>View source</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Number&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/Number"><span>Read</span></a></li><li id="ca-more-viewsource" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Number&action=edit"><span>View source</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Number&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/Number" 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/Number" 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="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Number&oldid=1256779505" 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=Number&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=Number&id=1256779505&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%2FNumber"><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%2FNumber"><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=Number&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=Number&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/Numbers" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://en.wikiquote.org/wiki/Number" 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/Q11563" 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-pp-default" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected."><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></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">Used to count, measure, and label</div> <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 other uses, see <a href="/wiki/Number_(disambiguation)" class="mw-disambig" title="Number (disambiguation)">Number (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NumberSetinC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/220px-NumberSetinC.svg.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/330px-NumberSetinC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/440px-NumberSetinC.svg.png 2x" data-file-width="600" data-file-height="470" /></a><figcaption><a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">Set inclusions</a> between the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> (ℕ), the <a href="/wiki/Integer" title="Integer">integers</a> (ℤ), the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (ℚ), the <a href="/wiki/Real_number" title="Real number">real numbers</a> (ℝ), and the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> (ℂ)</figcaption></figure> <p>A <b>number</b> is a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> used to count, measure, and label. The most basic examples are the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> 1, 2, 3, 4, and so forth.<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> Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called <i>numerals</i>; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a>, which is an organized way to represent any number. The most common numeral system is the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>, which allows for the representation of any <a href="/wiki/Integer" title="Integer">non-negative integer</a> using a combination of ten fundamental numeric symbols, called <a href="/wiki/Numerical_digit" title="Numerical digit">digits</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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with <a href="/wiki/Serial_number" title="Serial number">serial numbers</a>), and for codes (as with <a href="/wiki/ISBN" title="ISBN">ISBNs</a>). In common usage, a <i>numeral</i> is not clearly distinguished from the <i>number</i> that it represents. </p><p>In mathematics, the notion of number has been extended over the centuries to include zero (0),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>,<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> such as <a href="/wiki/One_half" title="One half">one half</a> <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 \left({\tfrac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {1}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71438ef4e12eecc24f98381504ddd12ad24a3905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.788ex; height:3.509ex;" alt="{\displaystyle \left({\tfrac {1}{2}}\right)}"></span>, <a href="/wiki/Real_number" title="Real number">real numbers</a> such as the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> <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 \left({\sqrt {2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd8558d246433e36e98f5079a7e014c1a5a2dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.228ex; height:3.343ex;" alt="{\displaystyle \left({\sqrt {2}}\right)}"></span> and <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Complex_number" title="Complex number">complex numbers</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> which extend the real numbers with a <a href="/wiki/Imaginary_unit" title="Imaginary unit">square root of <span class="texhtml">−1</span></a> (and its combinations with real numbers by adding or subtracting its multiples).<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Calculation" title="Calculation">Calculations</a> with numbers are done with arithmetical operations, the most familiar being <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>. Their study or usage is called <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, a term which may also refer to <a href="/wiki/Number_theory" title="Number theory">number theory</a>, the study of the properties of numbers. </p><p>Besides their practical uses, numbers have cultural significance throughout the world.<sup id="cite_ref-Gilsdorf_8-0" class="reference"><a href="#cite_note-Gilsdorf-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Restivo_9-0" class="reference"><a href="#cite_note-Restivo-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, in Western society, the <a href="/wiki/13_(number)" title="13 (number)">number 13</a> is often regarded as <a href="/wiki/Unlucky" class="mw-redirect" title="Unlucky">unlucky</a>, and "<a href="/wiki/One_million" class="mw-redirect" title="One million">a million</a>" may signify "a lot" rather than an exact quantity.<sup id="cite_ref-Gilsdorf_8-1" class="reference"><a href="#cite_note-Gilsdorf-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Though it is now regarded as <a href="/wiki/Pseudoscience" title="Pseudoscience">pseudoscience</a>, belief in a mystical significance of numbers, known as <a href="/wiki/Numerology" title="Numerology">numerology</a>, permeated ancient and medieval thought.<sup id="cite_ref-Ore_10-0" class="reference"><a href="#cite_note-Ore-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Numerology heavily influenced the development of <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a>, stimulating the investigation of many problems in number theory which are still of interest today.<sup id="cite_ref-Ore_10-1" class="reference"><a href="#cite_note-Ore-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex numbers</a>, which consist of various extensions or modifications of the <a href="/wiki/Complex_number" title="Complex number">complex number</a> system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> and <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, and the application of the term "number" is a matter of convention, without fundamental significance.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <div class="mw-heading mw-heading3"><h3 id="First_use_of_numbers">First use of numbers</h3></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/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History of ancient numeral systems</a></div> <p>Bones and other artifacts have been discovered with marks cut into them that many believe are <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. </p><p>A tallying system has no concept of place value (as in modern <a href="/wiki/Decimal" title="Decimal">decimal</a> notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system. </p><p>The first known system with place value was the <a href="/wiki/Ancient_Mesopotamian_units_of_measurement" title="Ancient Mesopotamian units of measurement">Mesopotamian base 60</a> system (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 3400</span> BC) and the earliest known base 10 system dates to 3100 BC in <a href="/wiki/Egypt" title="Egypt">Egypt</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Numerals">Numerals</h3></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>Numbers should be distinguished from <b>numerals</b>, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today.<sup id="cite_ref-Cengage_Learning2_15-0" class="reference"><a href="#cite_note-Cengage_Learning2-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="This claim needs references to better sources. (January 2017)">better source needed</span></a></i>]</sup> The key to the effectiveness of the system was the symbol for <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>, which was developed by ancient <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematicians</a> around 500 AD.<sup id="cite_ref-Cengage_Learning2_15-1" class="reference"><a href="#cite_note-Cengage_Learning2-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Zero">Zero<span class="anchor" id="History_of_zero"></span></h3></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Number" title="Special:EditPage/Number">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">November 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The first known documented use of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> dates to AD 628, and appeared in the <i><a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a></i>, the main work of the <a href="/wiki/Indian_mathematician" class="mw-redirect" title="Indian mathematician">Indian mathematician</a> <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>. He treated 0 as a number and discussed operations involving it, including <a href="/wiki/Division_by_zero" title="Division by zero">division</a>. By this time (the 7th century) the concept had clearly reached Cambodia as <a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer numerals</a>, and documentation shows the idea later spreading to China and the <a href="/wiki/Islamic_world" class="mw-redirect" title="Islamic world">Islamic world</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg/220px-Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg 1.5x" data-file-width="244" data-file-height="157" /></a><figcaption>The number 605 in <a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer numerals</a>, from an inscription from 683 AD. Early use of zero as a decimal figure.</figcaption></figure> <p>Brahmagupta's <i>Brāhmasphuṭasiddhānta</i> is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The <i>Brāhmasphuṭasiddhānta</i> is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. </p><p>The use of 0 as a number should be distinguished from its use as a placeholder numeral in <a href="/wiki/Place-value_system" class="mw-redirect" title="Place-value system">place-value systems</a>. Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word <i>nfr</i> to denote zero balance in <a href="/wiki/Double-entry_bookkeeping_system" class="mw-redirect" title="Double-entry bookkeeping system">double entry accounting</a>. Indian texts used a <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> word <span title="Sanskrit-language text"><i lang="sa-Latn">Shunye</i></span> or <span title="Sanskrit-language text"><i lang="sa">shunya</i></span> to refer to the concept of <i>void</i>. In mathematics texts this word often refers to the number zero.<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> In a similar vein, <a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a> (5th century BC) used the null (zero) operator in the <i><a href="/wiki/Ashtadhyayi" class="mw-redirect" title="Ashtadhyayi">Ashtadhyayi</a></i>, an early example of an <a href="/wiki/Formal_grammar" title="Formal grammar">algebraic grammar</a> for the Sanskrit language (also see <a href="/wiki/Pingala" title="Pingala">Pingala</a>). </p><p>There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the <i>Brāhmasphuṭasiddhānta</i>. </p><p>Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting <a href="/wiki/Philosophical" class="mw-redirect" title="Philosophical">philosophical</a> and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The <a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">paradoxes</a> of <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a> depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a> was a number.) </p><p>The late <a href="/wiki/Olmec" class="mw-redirect" title="Olmec">Olmec</a> people of south-central Mexico began to use a symbol for zero, a shell <a href="/wiki/Glyph" title="Glyph">glyph</a>, in the New World, possibly by the <span class="nowrap">4th century BC</span> but certainly by 40 BC, which became an integral part of <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a> and the <a href="/wiki/Maya_calendar" title="Maya calendar">Maya calendar</a>. Maya arithmetic used base 4 and base 5 written as base 20. <a href="/wiki/George_I._S%C3%A1nchez" title="George I. Sánchez">George I. Sánchez</a> in 1961 reported a base 4, base 5 "finger" abacus.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="The only source is a self-published book, albeit one by a respected educator. According to the (favorable) review by David H. Kelley in 'American Anthropologist', Sánchez was neither a Mayanist nor a mathematician. The review does not mention the abacus. (September 2020)">better source needed</span></a></i>]</sup> </p><p>By 130 AD, <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>, influenced by <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> numeral system otherwise using alphabetic <a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a>. Because it was used alone, not as just a placeholder, this <a href="/wiki/Greek_numerals#Hellenistic_zero" title="Greek numerals">Hellenistic zero</a> was the first <i>documented</i> use of a true zero in the Old World. In later <a href="/wiki/Byzantine_Empire" title="Byzantine Empire">Byzantine</a> manuscripts of his <i>Syntaxis Mathematica</i> (<i>Almagest</i>), the Hellenistic zero had morphed into the Greek letter <a href="/wiki/Omicron" title="Omicron">Omicron</a> (otherwise meaning 70). </p><p>Another true zero was used in tables alongside <a href="/wiki/Roman_numerals#Zero" title="Roman numerals">Roman numerals</a> by 525 (first known use by <a href="/wiki/Dionysius_Exiguus" title="Dionysius Exiguus">Dionysius Exiguus</a>), but as a word, <span title="Latin-language text"><i lang="la">nulla</i></span> meaning <i>nothing</i>, not as a symbol. When division produced 0 as a remainder, <span title="Latin-language text"><i lang="la">nihil</i></span>, also meaning <i>nothing</i>, was used. These medieval zeros were used by all future medieval <a href="/wiki/Computus" class="mw-redirect" title="Computus">computists</a> (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by <a href="/wiki/Bede" title="Bede">Bede</a> or a colleague about 725, a true zero symbol. </p> <div class="mw-heading mw-heading3"><h3 id="Negative_numbers">Negative numbers <span class="anchor" id="History_of_negative_numbers"></span></h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_negative_numbers" class="mw-redirect" title="History of negative numbers">History of negative numbers</a></div> <p>The abstract concept of negative numbers was recognized as early as 100–50 BC in China. <i><a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a></i> contains methods for finding the areas of figures; red rods were used to denote positive <a href="/wiki/Coefficient" title="Coefficient">coefficients</a>, black for negative.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The first reference in a Western work was in the 3rd century AD in Greece. <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> referred to the equation equivalent to <span class="nowrap">4<i>x</i> + 20 = 0</span> (the solution is negative) in <i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i>, saying that the equation gave an absurd result. </p><p>During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>, in <i><a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a></i> in 628, who used negative numbers to produce the general form <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a> that remains in use today. However, in the 12th century in India, <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhaskara</a> gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots". </p><p>European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of <span title="Latin-language text"><i lang="la"><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></span>, 1202) and later as losses (in <span title="Latin-language text"><i lang="la">Flos</i></span>). <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The first use of negative numbers in a European work was by <a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Nicolas Chuquet</a> during the 15th century. He used them as <a href="/wiki/Exponent" class="mw-redirect" title="Exponent">exponents</a>, but referred to them as "absurd numbers". </p><p>As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. </p> <div class="mw-heading mw-heading3"><h3 id="Rational_numbers">Rational numbers <span class="anchor" id="History_of_rational_numbers"></span></h3></div> <p>It is likely that the concept of fractional numbers dates to <a href="/wiki/Prehistoric_times" class="mw-redirect" title="Prehistoric times">prehistoric times</a>. The <a href="/wiki/Ancient_Egyptians" class="mw-redirect" title="Ancient Egyptians">Ancient Egyptians</a> used their <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</a> notation for rational numbers in mathematical texts such as the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> and the <a href="/wiki/Kahun_Papyrus" class="mw-redirect" title="Kahun Papyrus">Kahun Papyrus</a>. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of <a href="/wiki/Number_theory" title="Number theory">number theory</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The best known of these is <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, dating to roughly 300 BC. Of the Indian texts, the most relevant is the <a href="/wiki/Sthananga_Sutra" title="Sthananga Sutra">Sthananga Sutra</a>, which also covers number theory as part of a general study of mathematics. </p><p>The concept of <a href="/wiki/Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fractions</a> is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math <a href="/wiki/Sutra" title="Sutra">sutra</a> to include calculations of decimal-fraction approximations to <a href="/wiki/Pi" title="Pi">pi</a> or the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2020)">citation needed</span></a></i>]</sup> Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency. </p> <div class="mw-heading mw-heading3"><h3 id="Irrational_numbers">Irrational numbers <span class="anchor" id="History_of_irrational_numbers"></span></h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_irrational_numbers" class="mw-redirect" title="History of irrational numbers">History of irrational numbers</a></div> <p>The earliest known use of irrational numbers was in the <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a> <a href="/wiki/Sulba_Sutras" class="mw-redirect" title="Sulba Sutras">Sulba Sutras</a> composed between 800 and 500 BC.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="Source may be unreliable it garbles both the history and the mathematics. Source only says the mathematics in the Shulba Sutras "leads to the concept of irrational numbers". Since good approximations of irrational numbers appeared in earlier times, it's not clear what special role is being claimed for the Shulba Sutras in the history of irrational numbers. Also, should page reference be to p. 412 rather than p. 451? (September 2020)">better source needed</span></a></i>]</sup> The first existence proofs of irrational numbers is usually attributed to <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>, more specifically to the <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagorean</a> <a href="/wiki/Hippasus" title="Hippasus">Hippasus of Metapontum</a>, who produced a (most likely geometrical) proof of the irrationality of the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="Hippasus is mentioned only briefly in passing in this work. Entire books have been written on Pythagoras and Pythagoreanism; surely a reference could be provide to one of those? But any serious work will say that everything in this paragraph is unreliable myth, and some is outright modern fabrication, e.g. Pythagoras sentencing Hippasus to death. (September 2020)">better source needed</span></a></i>]</sup> </p><p>The 16th century brought final European acceptance of negative integral and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since <a href="/wiki/Euclid" title="Euclid">Euclid</a>. In 1872, the publication of the theories of <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> (by his pupil E. Kossak), <a href="/wiki/Eduard_Heine" title="Eduard Heine">Eduard Heine</a>,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> was brought about. In 1869, <a href="/wiki/Charles_M%C3%A9ray" title="Charles Méray">Charles Méray</a> had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by <a href="/wiki/Salvatore_Pincherle" title="Salvatore Pincherle">Salvatore Pincherle</a> (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by <a href="/wiki/Paul_Tannery" title="Paul Tannery">Paul Tannery</a> (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a <a href="/wiki/Dedekind_cut" title="Dedekind cut">cut (Schnitt)</a> in the system of <a href="/wiki/Real_number" title="Real number">real numbers</a>, separating all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a>,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and Méray. </p><p>The search for roots of <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic</a> and higher degree equations was an important development, the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> (<a href="/wiki/Paolo_Ruffini_(mathematician)" class="mw-redirect" title="Paolo Ruffini (mathematician)">Ruffini</a> 1799, <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Abel</a> 1824) showed that they could not be solved by <a href="/wiki/Nth_root" title="Nth root">radicals</a> (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of <a href="/wiki/Algebraic_numbers" class="mw-redirect" title="Algebraic numbers">algebraic numbers</a> (all solutions to polynomial equations). <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois</a> (1832) linked polynomial equations to <a href="/wiki/Group_theory" title="Group theory">group theory</a> giving rise to the field of <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>. </p><p><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a>, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and at the opening of the 19th century were brought into prominence through the writings of <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> first connected the subject with <a href="/wiki/Determinant" title="Determinant">determinants</a>, resulting, with the subsequent contributions of Heine,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a>, and Günther,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> in the theory of <span title="German-language text"><i lang="de">Kettenbruchdeterminanten</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Transcendental_numbers_and_reals">Transcendental numbers and reals <span class="anchor" id="History_of_transcendental_numbers_and_reals"></span></h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_%CF%80" class="mw-redirect" title="History of π">History of π</a></div> <p>The existence of <a href="/wiki/Transcendental_numbers" class="mw-redirect" title="Transcendental numbers">transcendental numbers</a><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> was first established by <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a> (1844, 1851). <a href="/wiki/Charles_Hermite" title="Charles Hermite">Hermite</a> proved in 1873 that <i>e</i> is transcendental and <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Lindemann</a> proved in 1882 that π is transcendental. Finally, <a href="/wiki/Cantor%27s_first_uncountability_proof" class="mw-redirect" title="Cantor's first uncountability proof">Cantor</a> showed that the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a> is <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountably infinite</a> but the set of all <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a> is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countably infinite</a>, so there is an uncountably infinite number of transcendental numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Infinity_and_infinitesimals">Infinity and infinitesimals <span class="anchor" id="History_of_infinity_and_infinitesimals"></span></h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_infinity" class="mw-redirect" title="History of infinity">History of infinity</a></div> <p>The earliest known conception of mathematical <a href="/wiki/Infinity" title="Infinity">infinity</a> appears in the <a href="/wiki/Yajur_Veda" class="mw-redirect" title="Yajur Veda">Yajur Veda</a>, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the <a href="/wiki/Jain" class="mw-redirect" title="Jain">Jain</a> mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. 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 {\text{∞}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>∞</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{∞}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50972cde37b4c1ef2bae09931ee75299fd198d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle {\text{∞}}}"></span> is often used to represent an infinite quantity. </p><p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> defined the traditional Western notion of mathematical infinity. He distinguished between <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a> and <a href="/wiki/Potential_infinity" class="mw-redirect" title="Potential infinity">potential infinity</a>—the general consensus being that only the latter had true value. <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>'s <i><a href="/wiki/Two_New_Sciences" title="Two New Sciences">Two New Sciences</a></i> discussed the idea of <a href="/wiki/Bijection" title="Bijection">one-to-one correspondences</a> between infinite sets. But the next major advance in the theory was made by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>; in 1895 he published a book about his new <a href="/wiki/Set_theory" title="Set theory">set theory</a>, introducing, among other things, <a href="/wiki/Transfinite_number" title="Transfinite number">transfinite numbers</a> and formulating the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>. </p><p>In the 1960s, <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a> showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of <a href="/wiki/Hyperreal_numbers" class="mw-redirect" title="Hyperreal numbers">hyperreal numbers</a> represents a rigorous method of treating the ideas about <a href="/wiki/Infinity" title="Infinity">infinite</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>. </p><p>A modern geometrical version of infinity is given by <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a> drawing. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers <span class="anchor" id="History_of_complex_numbers"></span></h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_complex_numbers" class="mw-redirect" title="History of complex numbers">History of complex numbers</a></div> <p>The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor <a href="/wiki/Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a> in the <span class="nowrap">1st century AD</span>, when he considered the volume of an impossible <a href="/wiki/Frustum" title="Frustum">frustum</a> of a <a href="/wiki/Pyramid" title="Pyramid">pyramid</a>. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> and <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. </p><p>This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary number</a> for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation </p> <dl><dd><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 \left({\sqrt {-1}}\right)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {-1}}\right)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be613a8ea2a3b2259994e0f2911ca3ee5cacb0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.071ex; height:3.676ex;" alt="{\displaystyle \left({\sqrt {-1}}\right)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}"></span></dd></dl> <p>seemed capriciously inconsistent with the algebraic identity </p> <dl><dd><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 {a}}{\sqrt {b}}={\sqrt {ab}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>b</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b49ce0866d6d7d45f4c67b19092573a3a1cd7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.008ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},}"></span></dd></dl> <p>which is valid for positive real numbers <i>a</i> and <i>b</i>, and was also used in complex number calculations with one of <i>a</i>, <i>b</i> positive and the other negative. The incorrect use of this identity, and the related identity </p> <dl><dd><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 {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>a</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d624c7d735f665ef45a9498e982f9f5c52d261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.49ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}"></span></dd></dl> <p>in the case when both <i>a</i> and <i>b</i> are negative even bedeviled <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> This difficulty eventually led him to the convention of using the special symbol <i>i</i> in place 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 {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}"></span> to guard against this mistake. </p><p>The 18th century saw the work of <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">De Moivre's formula</a> (1730) states: </p> <dl><dd><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 (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c64ddb01517848cd8e106cff324f7a88565703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.819ex; height:2.843ex;" alt="{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta }"></span></dd></dl> <p>while <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> (1748) gave us: </p> <dl><dd><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 \cos \theta +i\sin \theta =e^{i\theta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta +i\sin \theta =e^{i\theta }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56e70980172aa83a815bef9bdf4cc1c0016b853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.351ex; height:2.843ex;" alt="{\displaystyle \cos \theta +i\sin \theta =e^{i\theta }.}"></span></dd></dl> <p>The existence of complex numbers was not completely accepted until <a href="/wiki/Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a> described the geometrical interpretation in 1799. <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in <a href="/wiki/John_Wallis" title="John Wallis">Wallis</a>'s <i>De algebra tractatus</i>. </p><p>In the same year, Gauss provided the first generally accepted proof of the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>, showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of the form <span class="nowrap"><i>a</i> + <i>bi</i></span>, where <i>a</i> and <i>b</i> are integers (now called <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>) or rational numbers. His student, <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Gotthold Eisenstein</a>, studied the type <span class="nowrap"><i>a</i> + <i>bω</i></span>, where <i>ω</i> is a complex root of <span class="nowrap"><i>x</i><sup>3</sup> − 1 = 0</span> (now called <a href="/wiki/Eisenstein_integers" class="mw-redirect" title="Eisenstein integers">Eisenstein integers</a>). Other such classes (called <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic fields</a>) of complex numbers derive from the <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> <span class="nowrap"><i>x</i><sup><i>k</i></sup> − 1 = 0</span> for higher values of <i>k</i>. This generalization is largely due to <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>, who also invented <a href="/wiki/Ideal_number" title="Ideal number">ideal numbers</a>, which were expressed as geometrical entities by <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> in 1893. </p><p>In 1850 <a href="/wiki/Victor_Alexandre_Puiseux" class="mw-redirect" title="Victor Alexandre Puiseux">Victor Alexandre Puiseux</a> took the key step of distinguishing between poles and branch points, and introduced the concept of <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">essential singular points</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Why is this a key step in the history of complex numbers? (September 2020)">clarification needed</span></a></i>]</sup> This eventually led to the concept of the <a href="/wiki/Extended_complex_plane" class="mw-redirect" title="Extended complex plane">extended complex plane</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Prime_numbers">Prime numbers <span class="anchor" id="History_of_prime_numbers"></span></h3></div> <p><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> have been studied throughout recorded history.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Wikipedia's prime number article says the Greeks were the first to explicitly study prime numbers and mentions only the Rhind Papyrus as implicitly recognizing a distinction between prime and composite numbers. (September 2020)">citation needed</span></a></i>]</sup> They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of the <i>Elements</i> to the theory of primes; in it he proved the infinitude of the primes and the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, and presented the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> for finding the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of two numbers. </p><p>In 240 BC, <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a> used the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a> to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> and later eras.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Need citation for activity (or lack thereof) during era between Eratosthenes and Legendre. (September 2020)">citation needed</span></a></i>]</sup> </p><p>In 1796, <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> conjectured the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the <a href="/wiki/Goldbach_conjecture" class="mw-redirect" title="Goldbach conjecture">Goldbach conjecture</a>, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, formulated by <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> in 1859. The <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> was finally proved by <a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a> and <a href="/wiki/Charles_de_la_Vall%C3%A9e-Poussin" class="mw-redirect" title="Charles de la Vallée-Poussin">Charles de la Vallée-Poussin</a> in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted. </p> <div class="mw-heading mw-heading2"><h2 id="Main_classification">Main classification<span class="anchor" id="Classification"></span><span class="anchor" id="Classification_of_numbers"></span></h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Number system" redirects here. For systems which express numbers, see <a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List of types of numbers</a></div> <p>Numbers can be classified into <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, called <b>number sets</b> or <b>number systems</b>, such as the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> and the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>. The main number systems are as follows: </p> <table class="wikitable" style="margin: 1em auto; max-width: 600px; overflow-x: auto"> <caption>Main number systems </caption> <tbody><tr> <th>Symbol </th> <th>Name </th> <th>Examples/Explanation </th></tr> <tr> <th><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> </th> <th><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> </th> <td>0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...<br /> <p><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> or <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> are sometimes used. </p> </td></tr> <tr> <th><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> </th> <th><a href="/wiki/Integer" title="Integer">Integers</a> </th> <td>..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... </td></tr> <tr> <th><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> </th> <th><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> </th> <td><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span> where <i>a</i> and <i>b</i> are integers and <i>b</i> is not 0 </td></tr> <tr> <th><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> </th> <th><a href="/wiki/Real_number" title="Real number">Real numbers</a> </th> <td>The limit of a convergent sequence of rational numbers </td></tr> <tr> <th><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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span> </th> <th><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> </th> <td><i>a</i> + <i>bi</i> where <i>a</i> and <i>b</i> are real numbers and <i>i</i> is a formal square root of −1 </td></tr></tbody></table> <p>Each of these number systems is a <a href="/wiki/Subset" title="Subset">subset</a> of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as </p> <dl><dd><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} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579ab35e12fec7fdceb06b0085830426734b946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.787ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }"></span>.</dd></dl> <p>A more complete list of number sets appears in the following diagram. </p> <table style="margin:2em; border:2px solid silver; font-size:95%; border-collapse:collapse"> <tbody><tr> <td> <table style="margin:4px; border:2px solid silver"> <tbody><tr> <td> <table style="margin:1em"> <caption><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number systems</a> </caption> <tbody><tr> <td><a href="/wiki/Complex_number" title="Complex number">Complex</a> <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {C} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Real_number" title="Real number">Real</a> <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"> <mo>:</mo> <mspace width="thickmathspace" /> <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/9b09bba427588b2a529ebcf8fdb7536da42003b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {R} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Rational_number" title="Rational number">Rational</a> <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"> <mo>:</mo> <mspace width="thickmathspace" /> <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/9f77b368ade52a03084dad12fba5b25129cebe0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Integer" title="Integer">Integer</a> <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"> <mo>:</mo> <mspace width="thickmathspace" /> <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/cff631a0751189f28ca66b5d8ab161f05259f8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {Z} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Natural_number" title="Natural number">Natural</a> <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"> <mo>:</mo> <mspace width="thickmathspace" /> <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/51ba123110cb54a0b89909e10845ed2ee8c52e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>: 0 </td></tr> <tr> <td><a href="/wiki/One" class="mw-redirect" title="One">One</a>: 1 </td></tr> <tr> <td><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> </td></tr> <tr> <td><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Fraction" title="Fraction">Fraction</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a> </td></tr> <tr> <td><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a> </td></tr> <tr> <td><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> </td> <td> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Irrational_number" title="Irrational number">Irrational</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic irrational</a> </td></tr> <tr> <td><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a> </td></tr> <tr> <td><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Natural_numbers">Natural numbers</h3></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/Natural_number" title="Natural number">Natural number</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Nat_num.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/220px-Nat_num.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/330px-Nat_num.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/440px-Nat_num.svg.png 2x" data-file-width="120" data-file-height="120" /></a><figcaption>The natural numbers, starting with 1</figcaption></figure> <p>The most familiar numbers are the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, <a href="/wiki/Set_theory" title="Set theory">set theorists</a> and other mathematicians started including 0 (<a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, i.e. 0 elements, where 0 is thus the smallest <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>) in the set of natural numbers.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Today, different mathematicians use the term to describe both sets, including 0 or not. The <a href="/wiki/Mathematical_symbol" class="mw-redirect" title="Mathematical symbol">mathematical symbol</a> for the set of all natural numbers is <b>N</b>, also 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 \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>, and sometimes <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> or <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> when it is necessary to indicate whether the set should start with 0 or 1, respectively. </p><p>In the <a href="/wiki/Base_10" class="mw-redirect" title="Base 10">base 10</a> numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a>: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The <a href="/wiki/Radix" title="Radix">radix or base</a> is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base 10 system, the rightmost digit of a natural number has a <a href="/wiki/Place_value" class="mw-redirect" title="Place value">place value</a> of 1, and every other digit has a place value ten times that of the place value of the digit to its right. </p><p>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, which is capable of acting as an axiomatic foundation for modern mathematics,<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in <a href="/wiki/Peano_Arithmetic" class="mw-redirect" title="Peano Arithmetic">Peano Arithmetic</a>, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times. </p> <div class="mw-heading mw-heading3"><h3 id="Integers">Integers</h3></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/Integer" title="Integer">Integer</a></div> <p>The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a <a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a>). As an example, the negative of 7 is written −7, and <span class="nowrap">7 + (−7) = 0</span>. When the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of <a href="/wiki/Integer" title="Integer">integers</a>, <b>Z</b> also written <a href="/wiki/Blackboard_bold" title="Blackboard bold"><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></a>. Here the letter Z comes from <a href="/wiki/German_language" title="German language">German</a> <i> Zahl</i> 'number'. The set of integers forms a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> with the operations addition and multiplication.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as <b>positive integers</b>, and the natural numbers with zero are referred to as <b>non-negative integers</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Rational_numbers_2">Rational numbers</h3></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/Rational_number" title="Rational number">Rational number</a></div> <p>A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>m</i></span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span> represents <i>m</i> parts of a whole divided into <i>n</i> equal parts. Two different fractions may correspond to the same rational number; for example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> are equal, that is: </p> <dl><dd><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 \over 2}={2 \over 4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over 2}={2 \over 4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b6a5243bcf697d6f4e39f8102e66e3888269b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.742ex; height:5.176ex;" alt="{\displaystyle {1 \over 2}={2 \over 4}.}"></span></dd></dl> <p>In general, </p> <dl><dd><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 \over b}={c \over d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a \over b}={c \over d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0696d2d2f38d22a180747bc53adbd900b638ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.216ex; height:4.843ex;" alt="{\displaystyle {a \over b}={c \over d}}"></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 {a\times d}={c\times b}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>×</mo> <mi>d</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>×</mo> <mi>b</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a\times d}={c\times b}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3fe0d5559d25ae52e0661aa99eee495c1554a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.876ex; height:2.176ex;" alt="{\displaystyle {a\times d}={c\times b}.}"></span></dd></dl> <p>If the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of <i>m</i> is greater than <i>n</i> (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example −7 can be written <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">−7</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span>. The symbol for the rational numbers is <b>Q</b> (for <i><a href="/wiki/Quotient" title="Quotient">quotient</a></i>), also written <a href="/wiki/Blackboard_bold" title="Blackboard bold"><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>.</a> </p> <div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3></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/Real_number" title="Real number">Real number</a></div> <p>The symbol for the real numbers is <b>R</b>, also 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 \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> <mo>.</mo> </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/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"></span> They include all the measuring numbers. Every real number corresponds to a point on the <a href="/wiki/Number_line" title="Number line">number line</a>. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a <a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a>, e.g. −123.456. </p><p>Most real numbers can only be <i>approximated</i> by <a href="/wiki/Decimal" title="Decimal">decimal</a> numerals, in which a <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, 123.456 represents <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">123456</span><span class="sr-only">/</span><span class="den">1000</span></span>⁠</span>, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its <a href="/wiki/Fractional_part" title="Fractional part">fractional part</a> has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a>. Thus <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.<span style="text-decoration:overline;">3</span>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>It turns out that these repeating decimals (including the <a href="/wiki/Trailing_zero" title="Trailing zero">repetition of zeroes</a>) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>. A famous irrational real number is the <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>, the ratio of the <a href="/wiki/Circumference" title="Circumference">circumference</a> of any circle to its <a href="/wiki/Diameter" title="Diameter">diameter</a>. When pi is written as </p> <dl><dd><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 \pi =3.14159265358979\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3.14159265358979</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3.14159265358979\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789b1a70d48edb3b613ffa907f5bed42539c7f74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.658ex; height:2.509ex;" alt="{\displaystyle \pi =3.14159265358979\dots ,}"></span></dd></dl> <p>as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that <a href="/wiki/Proof_that_pi_is_irrational" class="mw-redirect" title="Proof that pi is irrational"><span class="texhtml mvar" style="font-style:italic;">π</span> is irrational</a>. Another well-known number, proven to be an irrational real number, is </p> <dl><dd><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}}=1.41421356237\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mn>1.41421356237</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=1.41421356237\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e5c092bd8a67d1e0d3eb894905af355aaa1bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.937ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}=1.41421356237\dots ,}"></span></dd></dl> <p>the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions <span class="nowrap">( 1 trillion = 10<sup>12</sup> = 1,000,000,000,000 )</span> of digits. </p><p>Not only these prominent examples but <a href="/wiki/Almost_all" title="Almost all">almost all</a> real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting <a href="/wiki/Rounding" title="Rounding">rounded</a> or <a href="/wiki/Truncation" title="Truncation">truncated</a> real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only <a href="/wiki/Countably_many" class="mw-redirect" title="Countably many">countably many</a>. All measurements are, by their nature, approximations, and always have a <a href="/wiki/Margin_of_error" title="Margin of error">margin of error</a>. Thus 123.456 is considered an approximation of any real number greater or equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1234555</span><span class="sr-only">/</span><span class="den">10000</span></span>⁠</span> and strictly less than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1234565</span><span class="sr-only">/</span><span class="den">10000</span></span>⁠</span> (rounding to 3 decimals), or of any real number greater or equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">123456</span><span class="sr-only">/</span><span class="den">1000</span></span>⁠</span> and strictly less than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">123457</span><span class="sr-only">/</span><span class="den">1000</span></span>⁠</span> (truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called <a href="/wiki/Significant_digits" class="mw-redirect" title="Significant digits">significant digits</a>. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 <a href="/wiki/Metre" title="Metre">m</a>. If the sides of a rectangle are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between <span class="nowrap">5.614591 m<sup>2</sup></span> and <span class="nowrap">5.603011 m<sup>2</sup></span>. Since not even the second digit after the decimal place is preserved, the following digits are not <i>significant</i>. Therefore, the result is usually rounded to 5.61. </p><p>Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example, <a href="/wiki/0.999..." title="0.999...">0.999...</a>, 1.0, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9s, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000.... Similarly, a decimal numeral with an unlimited number of 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9s can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. Finally, an unlimited sequence of 0s to the right of a decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9s, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example, 99.999... = 100. </p><p>The real numbers also have an important but highly technical property called the <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a> property. </p><p>It can be shown that any <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a>, which is also <a href="/wiki/Completeness_of_the_real_numbers" title="Completeness of the real numbers">complete</a>, is isomorphic to the real numbers. The real numbers are not, however, an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>, because they do not include a solution (often called a <a href="/wiki/Square_root_of_minus_one" class="mw-redirect" title="Square root of minus one">square root of minus one</a>) to the algebraic 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 x^{2}+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers_2">Complex numbers</h3></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/Complex_number" title="Complex number">Complex number</a></div> <p>Moving to a greater level of abstraction, the real numbers can be extended to the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. This set of numbers arose historically from trying to find closed formulas for the roots of <a href="/wiki/Cubic_function" title="Cubic function">cubic</a> and <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic</a> polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a <a href="/wiki/Square_root" title="Square root">square root</a> of −1, denoted by <i><a href="/wiki/Imaginary_unit" title="Imaginary unit">i</a></i>, a symbol assigned by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, and called the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. The complex numbers consist of all numbers of the form </p> <dl><dd><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+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,a+bi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ccd15fa872e18fbaccb8c42fb43aef1d017dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.257ex; height:2.343ex;" alt="{\displaystyle \,a+bi}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are real numbers. Because of this, complex numbers correspond to points on the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, a <a href="/wiki/Vector_space" title="Vector space">vector space</a> of two real <a href="/wiki/Dimension" title="Dimension">dimensions</a>. In the expression <span class="nowrap"><i>a</i> + <i>bi</i></span>, the real number <i>a</i> is called the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> and <i>b</i> is called the <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a>. If the real part of a complex number is 0, then the number is called an <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary number</a> or is referred to as <i>purely imaginary</i>; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a <a href="/wiki/Subset" title="Subset">subset</a> of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a>. The symbol for the complex numbers is <b>C</b> or <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>. </p><p>The <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that the complex numbers form an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>, meaning that every <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with complex coefficients has a <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> in the complex numbers. Like the reals, the complex numbers form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, which is <a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">complete</a>, but unlike the real numbers, it is not <a href="/wiki/Total_order" title="Total order">ordered</a>. That is, there is no consistent meaning assignable to saying that <i>i</i> is greater than 1, nor is there any meaning in saying that <i>i</i> is less than 1. In technical terms, the complex numbers lack a <a href="/wiki/Total_order" title="Total order">total order</a> that is <a href="/wiki/Ordered_field" title="Ordered field">compatible with field operations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Subclasses_of_the_integers">Subclasses of the integers</h2></div> <div class="mw-heading mw-heading3"><h3 id="Even_and_odd_numbers">Even and odd numbers</h3></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/Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">Even and odd numbers</a></div> <p>An <b>even number</b> is an integer that is "evenly divisible" by two, that is <a href="/wiki/Euclidean_division" title="Euclidean division">divisible by two without remainder</a>; an <b>odd number</b> is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "<a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisible</a>".) Any odd number <i>n</i> may be constructed by the formula <span class="nowrap"><i>n</i> = 2<i>k</i> + 1,</span> for a suitable integer <i>k</i>. Starting with <span class="nowrap"><i>k</i> = 0,</span> the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number <i>m</i> has the form <span class="nowrap"><i>m</i> = 2<i>k</i></span> where <i>k</i> is again an <a href="/wiki/Integer" title="Integer">integer</a>. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}. </p> <div class="mw-heading mw-heading3"><h3 id="Prime_numbers_2">Prime numbers</h3></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/Prime_number" title="Prime number">Prime number</a></div> <p>A <b>prime number</b>, often shortened to just <b>prime</b>, is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to <a href="/wiki/Number_theory" title="Number theory">number theory</a>. <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> is an example of a still unanswered question: "Is every even number the sum of two primes?" </p><p>One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. A proof appears in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_classes_of_integers">Other classes of integers</h3></div> <p>Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> and <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a>. For more examples, see <a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequence</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Subclasses_of_the_complex_numbers">Subclasses of the complex numbers</h2></div> <div class="mw-heading mw-heading3"><h3 id="Algebraic,_irrational_and_transcendental_numbers"><span id="Algebraic.2C_irrational_and_transcendental_numbers"></span>Algebraic, irrational and transcendental numbers</h3></div> <p><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>. Complex numbers which are not algebraic are called <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>. The algebraic numbers that are solutions of a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> equation with integer coefficients are called <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Periods_and_exponential_periods">Periods and exponential periods</h3></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/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Period (algebraic geometry)</a></div> <p>A period is a complex number that can be expressed as an <a href="/wiki/Integral" title="Integral">integral</a> of an <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> over an algebraic <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constants</a> such as the <a href="/wiki/Pi" title="Pi">number <i>π</i></a>. The set of periods form a countable <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> and bridge the gap between algebraic and transcendental numbers.<sup id="cite_ref-:1_38-0" class="reference"><a href="#cite_note-:1-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>The periods can be extended by permitting the integrand to be the product of an algebraic function and the <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> of an algebraic function. This gives another countable ring: the exponential periods. The <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">number <i>e</i></a> as well as <a href="/wiki/Euler%27s_constant" title="Euler's constant">Euler's constant</a> are exponential periods.<sup id="cite_ref-:1_38-1" class="reference"><a href="#cite_note-:1-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Constructible_numbers">Constructible numbers</h3></div> <p>Motivated by the classical problems of <a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">constructions with straightedge and compass</a>, the <a href="/wiki/Constructible_number" title="Constructible number">constructible numbers</a> are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps. </p> <div class="mw-heading mw-heading3"><h3 id="Computable_numbers">Computable numbers</h3></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/Computable_number" title="Computable number">Computable number</a></div> <p>A <b>computable number</b>, also known as <i>recursive number</i>, is a <a href="/wiki/Real_number" title="Real number">real number</a> such that there exists an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> which, given a positive number <i>n</i> as input, produces the first <i>n</i> digits of the computable number's decimal representation. Equivalent definitions can be given using <a href="/wiki/%CE%9C-recursive_function" class="mw-redirect" title="Μ-recursive function">μ-recursive functions</a>, <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a> or <a href="/wiki/%CE%9B-calculus" class="mw-redirect" title="Λ-calculus">λ-calculus</a>. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, and thus form a <a href="/wiki/Real_closed_field" title="Real closed field">real closed field</a> that contains the real <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>. </p><p>The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. </p><p>The set of computable numbers has the same cardinality as the natural numbers. Therefore, <a href="/wiki/Almost_all" title="Almost all">almost all</a> real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions_of_the_concept">Extensions of the concept</h2></div> <div class="mw-heading mw-heading3"><h3 id="p-adic_numbers"><i>p</i>-adic numbers</h3></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/P-adic_number" title="P-adic number"><i>p</i>-adic number</a></div> <p>The <i>p</i>-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what <a href="/wiki/Radix" title="Radix">base</a> is used for the digits: any base is possible, but a <a href="/wiki/Prime_number" title="Prime number">prime number</a> base provides the best mathematical properties. The set of the <i>p</i>-adic numbers contains the rational numbers, but is not contained in the complex numbers. </p><p>The elements of an <a href="/wiki/Algebraic_function_field" title="Algebraic function field">algebraic function field</a> over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> and algebraic numbers have many similar properties (see <a href="/wiki/Function_field_analogy" class="mw-redirect" title="Function field analogy">Function field analogy</a>). Therefore, they are often regarded as numbers by number theorists. The <i>p</i>-adic numbers play an important role in this analogy. </p> <div class="mw-heading mw-heading3"><h3 id="Hypercomplex_numbers">Hypercomplex numbers</h3></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/Hypercomplex_number" title="Hypercomplex number">hypercomplex number</a></div> <p>Some number systems that are not included in the complex numbers may be constructed from the real 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 \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> in a way that generalize the construction of the complex numbers. They are sometimes called <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex numbers</a>. They include the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> <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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>, introduced by Sir <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>, in which multiplication is not <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, the <a href="/wiki/Octonion" title="Octonion">octonions</a> <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 {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>, in which multiplication is not <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> in addition to not being commutative, and the <a href="/wiki/Sedenion" title="Sedenion">sedenions</a> <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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>, in which multiplication is not <a href="/wiki/Alternative_algebra" title="Alternative algebra">alternative</a>, neither associative nor commutative. The hypercomplex numbers include one real unit together with <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^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"></span> imaginary units, for which <i>n</i> is a non-negative integer. For example, quaternions can generally represented using the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>d</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03573028039c479ff446bafa5d31dc06aaf47dea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.749ex; height:2.509ex;" alt="{\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}"></span> </p><p>where the coefficients <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>, <span class="texhtml mvar" style="font-style:italic;">d</span> are real numbers, and <span class="texhtml"><b>i</b>, <b>j</b></span>, <span class="texhtml"><b>k</b></span> are 3 different imaginary units. </p><p>Each hypercomplex number system is a <a href="/wiki/Subset" title="Subset">subset</a> of the next hypercomplex number system of double dimensions obtained via the <a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a>. For example, the 4-dimensional quaternions <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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span> are a subset of the 8-dimensional quaternions <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 {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>, which are in turn a subset of the 16-dimensional sedenions <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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>, in turn a subset of the 32-dimensional <a href="/wiki/Trigintaduonion" title="Trigintaduonion">trigintaduonions</a> <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 {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" 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 {T} }"></span>, and <i><a href="/wiki/Ad_infinitum" title="Ad infinitum">ad infinitum</a></i> with <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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span> dimensions, with <i>n</i> being any non-negative integer. Including the complex and real numbers and their subsets, this can be expressed symbolically as: </p> <dl><dd><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} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \mathbb {O} \subset \mathbb {S} \subset \mathbb {T} \subset \cdots }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mo>⊂</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \mathbb {O} \subset \mathbb {S} \subset \mathbb {T} \subset \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbd270396192ad9799039fa2efc6c5f649b44e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.461ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \mathbb {O} \subset \mathbb {S} \subset \mathbb {T} \subset \cdots }"></span></dd></dl> <p>Alternatively, starting from the real 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 \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>, which have zero complex units, this can be expressed as </p> <dl><dd><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 {\mathcal {C}}_{0}\subset {\mathcal {C}}_{1}\subset {\mathcal {C}}_{2}\subset {\mathcal {C}}_{3}\subset {\mathcal {C}}_{4}\subset {\mathcal {C}}_{5}\subset \cdots \subset C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{0}\subset {\mathcal {C}}_{1}\subset {\mathcal {C}}_{2}\subset {\mathcal {C}}_{3}\subset {\mathcal {C}}_{4}\subset {\mathcal {C}}_{5}\subset \cdots \subset C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddfc1bc55f8e06fa3f3669624d7fc97966b48d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.969ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{0}\subset {\mathcal {C}}_{1}\subset {\mathcal {C}}_{2}\subset {\mathcal {C}}_{3}\subset {\mathcal {C}}_{4}\subset {\mathcal {C}}_{5}\subset \cdots \subset C_{n}}"></span></dd></dl> <p>with <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span> containing <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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span> dimensions.<sup id="cite_ref-Saniga_41-0" class="reference"><a href="#cite_note-Saniga-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transfinite_numbers">Transfinite numbers</h3></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/Transfinite_number" title="Transfinite number">transfinite number</a></div> <p>For dealing with infinite <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, the natural numbers have been generalized to the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a> and to the <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number. </p> <div class="mw-heading mw-heading3"><h3 id="Nonstandard_numbers">Nonstandard numbers</h3></div> <p><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a> are used in <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">non-standard analysis</a>. The hyperreals, or nonstandard reals (usually denoted as *<b>R</b>), denote an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> that is a proper <a href="/wiki/Field_extension" title="Field extension">extension</a> of the ordered field of <a href="/wiki/Real_number" title="Real number">real numbers</a> <b>R</b> and satisfies the <a href="/wiki/Transfer_principle" title="Transfer principle">transfer principle</a>. This principle allows true <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> statements about <b>R</b> to be reinterpreted as true first-order statements about *<b>R</b>. </p><p><a href="/wiki/Superreal_number" title="Superreal number">Superreal</a> and <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a> extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.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{clear:left;float:left;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"><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> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/Concrete_number" title="Concrete number">Concrete number</a></li> <li><a href="/wiki/List_of_numbers" title="List of numbers">List of numbers</a></li> <li><a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List of types of numbers</a></li> <li><a href="/wiki/Mathematical_constant" title="Mathematical constant">Mathematical constant</a> – Fixed number that has received a name</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></li> <li><a href="/wiki/Numerical_cognition" title="Numerical cognition">Numerical cognition</a></li> <li><a href="/wiki/Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">Orders of magnitude</a></li> <li><a href="/wiki/Physical_constant" title="Physical constant">Physical constant</a> – Universal and unchanging physical quantity</li> <li><a href="/wiki/Physical_quantity" title="Physical quantity">Physical quantity</a> – Measurable property of a material or system</li> <li><a href="/wiki/Pi" title="Pi">Pi</a> – Number, approximately 3.14</li> <li><a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a> – Method for representing or encoding numbers</li> <li><a href="/wiki/Prime_number" title="Prime number">Prime number</a> – Number divisible only by 1 or itself</li> <li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar (mathematics)</a> – Elements of a field, e.g. real numbers, in the context of linear algebra</li> <li><a href="/wiki/Subitizing_and_counting" class="mw-redirect" title="Subitizing and counting">Subitizing and counting</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">In <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, a <a href="/wiki/Numeral_(linguistics)" title="Numeral (linguistics)">numeral</a> can refer to a symbol like 5, but also to a word or a phrase that names a number, like "five hundred"; numerals include also other words representing numbers, like "dozen".</span> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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: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 class="citation journal cs1"><a rel="nofollow" class="external text" href="http://www.oed.com/view/Entry/129082">"number, n."</a> <i>OED Online</i>. Oxford University Press. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181004081907/http://www.oed.com/view/Entry/129082">Archived</a> from the original on 4 October 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=OED+Online&rft.atitle=number%2C+n.&rft_id=http%3A%2F%2Fwww.oed.com%2Fview%2FEntry%2F129082&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://www.oed.com/view/Entry/129111">"numeral, adj. and n."</a> <i>OED Online</i>. Oxford University Press. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220730095156/https://www.oed.com/start;jsessionid=B9929F0647C8EE5D4FDB3A3C1B2CA3C3?authRejection=true&url=%2Fview%2FEntry%2F129111">Archived</a> from the original on 30 July 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=OED+Online&rft.atitle=numeral%2C+adj.+and+n.&rft_id=http%3A%2F%2Fwww.oed.com%2Fview%2FEntry%2F129111&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFMatson" class="citation news cs1">Matson, John. <a rel="nofollow" class="external text" href="https://www.scientificamerican.com/article/history-of-zero/">"The Origin of Zero"</a>. <i>Scientific American</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170826235655/https://www.scientificamerican.com/article/history-of-zero/">Archived</a> from the original on 26 August 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=The+Origin+of+Zero&rft.aulast=Matson&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.scientificamerican.com%2Farticle%2Fhistory-of-zero%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgkin2005" class="citation book cs1">Hodgkin, Luke (2 June 2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88"><i>A History of Mathematics: From Mesopotamia to Modernity</i></a>. OUP Oxford. pp. 85–88. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-152383-0" title="Special:BookSources/978-0-19-152383-0"><bdi>978-0-19-152383-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190204012433/https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88">Archived</a> from the original on 4 February 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2017</span>.</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.pages=85-88&rft.pub=OUP+Oxford&rft.date=2005-06-02&rft.isbn=978-0-19-152383-0&rft.aulast=Hodgkin&rft.aufirst=Luke&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df6HlhlBuQUgC%26pg%3DPA88&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Mathematics across cultures : the history of non-western mathematics</i>. Dordrecht: Kluwer Academic. 2000. pp. 410–411. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0260-2" title="Special:BookSources/1-4020-0260-2"><bdi>1-4020-0260-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+across+cultures+%3A+the+history+of+non-western+mathematics&rft.place=Dordrecht&rft.pages=410-411&rft.pub=Kluwer+Academic&rft.date=2000&rft.isbn=1-4020-0260-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFDescartes1954" class="citation book cs1"><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a> (1954) [1637]. <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofrenede00rend"><i>La Géométrie: The Geometry of René Descartes with a facsimile of the first edition</i></a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60068-8" title="Special:BookSources/0-486-60068-8"><bdi>0-486-60068-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">20 April</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=La+G%C3%A9om%C3%A9trie%3A+The+Geometry+of+Ren%C3%A9+Descartes+with+a+facsimile+of+the+first+edition&rft.pub=Dover+Publications&rft.date=1954&rft.isbn=0-486-60068-8&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofrenede00rend&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-Gilsdorf-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gilsdorf_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gilsdorf_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilsdorf2012" class="citation book cs1">Gilsdorf, Thomas E. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IN8El-TTlSQC"><i>Introduction to cultural mathematics : with case studies in the Otomies and the Incas</i></a>. Hoboken, N.J.: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-19416-4" title="Special:BookSources/978-1-118-19416-4"><bdi>978-1-118-19416-4</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/793103475">793103475</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+cultural+mathematics+%3A+with+case+studies+in+the+Otomies+and+the+Incas&rft.place=Hoboken%2C+N.J.&rft.pub=Wiley&rft.date=2012&rft_id=info%3Aoclcnum%2F793103475&rft.isbn=978-1-118-19416-4&rft.aulast=Gilsdorf&rft.aufirst=Thomas+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIN8El-TTlSQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-Restivo-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Restivo_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRestivo1992" class="citation book cs1">Restivo, Sal P. (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V0RuCQAAQBAJ&q=Mathematics+in+Society+and+History"><i>Mathematics in society and history : sociological inquiries</i></a>. Dordrecht. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-011-2944-2" title="Special:BookSources/978-94-011-2944-2"><bdi>978-94-011-2944-2</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/883391697">883391697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+society+and+history+%3A+sociological+inquiries&rft.place=Dordrecht&rft.date=1992&rft_id=info%3Aoclcnum%2F883391697&rft.isbn=978-94-011-2944-2&rft.aulast=Restivo&rft.aufirst=Sal+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DV0RuCQAAQBAJ%26q%3DMathematics%2Bin%2BSociety%2Band%2BHistory&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-Ore-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ore_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ore_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1988" class="citation book cs1">Ore, Øystein (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Sl_6BPp7S0AC"><i>Number theory and its history</i></a>. New York: Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65620-9" title="Special:BookSources/0-486-65620-9"><bdi>0-486-65620-9</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/17413345">17413345</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.place=New+York&rft.pub=Dover&rft.date=1988&rft_id=info%3Aoclcnum%2F17413345&rft.isbn=0-486-65620-9&rft.aulast=Ore&rft.aufirst=%C3%98ystein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSl_6BPp7S0AC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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">Gouvêa, Fernando Q. <i><a href="/wiki/The_Princeton_Companion_to_Mathematics" title="The Princeton Companion to Mathematics">The Princeton Companion to Mathematics</a>, Chapter II.1, "The Origins of Modern Mathematics"</i>, p. 82. Princeton University Press, September 28, 2008. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2">978-0-691-11880-2</a>. "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the <i>p</i>-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."</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:r1238218222"><cite id="CITEREFMarshack1971" class="citation book cs1">Marshack, Alexander (1971). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vbQ9AAAAIAAJ"><i>The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation</i></a> ([1st ed.] ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-040535-2" title="Special:BookSources/0-07-040535-2"><bdi>0-07-040535-2</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/257105">257105</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+roots+of+civilization%3B+the+cognitive+beginnings+of+man%27s+first+art%2C+symbol%2C+and+notation.&rft.place=New+York&rft.edition=%5B1st+ed.%5D&rft.pub=McGraw-Hill&rft.date=1971&rft_id=info%3Aoclcnum%2F257105&rft.isbn=0-07-040535-2&rft.aulast=Marshack&rft.aufirst=Alexander&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvbQ9AAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin">"Egyptian Mathematical Papyri – Mathematicians of the African Diaspora"</a>. Math.buffalo.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150407231917/http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin">Archived</a> from the original on 7 April 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">30 January</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Egyptian+Mathematical+Papyri+%E2%80%93+Mathematicians+of+the+African+Diaspora&rft.pub=Math.buffalo.edu&rft_id=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Africa%2Fmad_ancient_egyptpapyrus.html%23berlin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChrisomalis2003" class="citation journal cs1">Chrisomalis, Stephen (1 September 2003). "The Egyptian origin of the Greek alphabetic numerals". <i>Antiquity</i>. <b>77</b> (297): 485–96. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0003598X00092541">10.1017/S0003598X00092541</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-598X">0003-598X</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:160523072">160523072</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Antiquity&rft.atitle=The+Egyptian+origin+of+the+Greek+alphabetic+numerals&rft.volume=77&rft.issue=297&rft.pages=485-96&rft.date=2003-09-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A160523072%23id-name%3DS2CID&rft.issn=0003-598X&rft_id=info%3Adoi%2F10.1017%2FS0003598X00092541&rft.aulast=Chrisomalis&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-Cengage_Learning2-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cengage_Learning2_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cengage_Learning2_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBullietCrossleyHeadrickHirsch2010" class="citation book cs1">Bulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192"><i>The Earth and Its Peoples: A Global History, Volume 1</i></a>. Cengage Learning. p. 192. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4390-8474-8" title="Special:BookSources/978-1-4390-8474-8"><bdi>978-1-4390-8474-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170128072424/https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192">Archived</a> from the original on 28 January 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">16 May</span> 2017</span>. <q>Indian mathematicians invented the concept of zero and developed the "Arabic" numerals and system of place-value notation used in most parts of the world today</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Earth+and+Its+Peoples%3A+A+Global+History%2C+Volume+1&rft.pages=192&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=978-1-4390-8474-8&rft.aulast=Bulliet&rft.aufirst=Richard&rft.au=Crossley%2C+Pamela&rft.au=Headrick%2C+Daniel&rft.au=Hirsch%2C+Steven&rft.au=Johnson%2C+Lyman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdOxl71w-jHEC%26pg%3DPA192&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120112073735/http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html">"Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question"</a>. Sunsite.utk.edu. 26 April 1999. Archived from <a rel="nofollow" class="external text" href="http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html">the original</a> on 12 January 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">30 January</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Historia+Matematica+Mailing+List+Archive%3A+Re%3A+%5BHM%5D+The+Zero+Story%3A+a+question&rft.pub=Sunsite.utk.edu&rft.date=1999-04-26&rft_id=http%3A%2F%2Fsunsite.utk.edu%2Fmath_archives%2F.http%2Fhypermail%2Fhistoria%2Fapr99%2F0197.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSánchez1961" class="citation book cs1"><a href="/wiki/George_I._S%C3%A1nchez" title="George I. Sánchez">Sánchez, George I.</a> (1961). <i>Arithmetic in Maya</i>. Austin, Texas: self published.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetic+in+Maya&rft.place=Austin%2C+Texas&rft.pub=self+published&rft.date=1961&rft.aulast=S%C3%A1nchez&rft.aufirst=George+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFStaszkowRobert_Bradshaw2004" class="citation book cs1">Staszkow, Ronald; Robert Bradshaw (2004). <i>The Mathematical Palette (3rd ed.)</i>. Brooks Cole. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-40365-4" title="Special:BookSources/0-534-40365-4"><bdi>0-534-40365-4</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+Mathematical+Palette+%283rd+ed.%29&rft.pages=41&rft.pub=Brooks+Cole&rft.date=2004&rft.isbn=0-534-40365-4&rft.aulast=Staszkow&rft.aufirst=Ronald&rft.au=Robert+Bradshaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFSmith1958" class="citation book cs1"><a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David Eugene</a> (1958). <i>History of Modern Mathematics</i>. Dover Publications. p. 259. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-20429-4" title="Special:BookSources/0-486-20429-4"><bdi>0-486-20429-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Modern+Mathematics&rft.pages=259&rft.pub=Dover+Publications&rft.date=1958&rft.isbn=0-486-20429-4&rft.aulast=Smith&rft.aufirst=David+Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/humanities/world-history/ancient-medieval/classical-greece/a/greek-culture">"Classical Greek culture (article)"</a>. <i>Khan Academy</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220504133917/https://www.khanacademy.org/humanities/world-history/ancient-medieval/classical-greece/a/greek-culture">Archived</a> from the original on 4 May 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">4 May</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khan+Academy&rft.atitle=Classical+Greek+culture+%28article%29&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fhumanities%2Fworld-history%2Fancient-medieval%2Fclassical-greece%2Fa%2Fgreek-culture&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFSelin2000" class="citation book cs1"><a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a>, ed. (2000). <i>Mathematics across cultures: the history of non-Western mathematics</i>. Kluwer Academic Publishers. p. 451. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-6481-3" title="Special:BookSources/0-7923-6481-3"><bdi>0-7923-6481-3</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+across+cultures%3A+the+history+of+non-Western+mathematics&rft.pages=451&rft.pub=Kluwer+Academic+Publishers&rft.date=2000&rft.isbn=0-7923-6481-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFBernard_Frischer1984" class="citation book cs1">Bernard Frischer (1984). "Horace and the Monuments: A New Interpretation of the Archytas <i>Ode</i>". In <a href="/wiki/D._R._Shackleton_Bailey" title="D. R. Shackleton Bailey">D.R. Shackleton Bailey</a> (ed.). <i>Harvard Studies in Classical Philology</i>. Harvard University Press. p. 83. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-37935-7" title="Special:BookSources/0-674-37935-7"><bdi>0-674-37935-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Horace+and+the+Monuments%3A+A+New+Interpretation+of+the+Archytas+Ode&rft.btitle=Harvard+Studies+in+Classical+Philology&rft.pages=83&rft.pub=Harvard+University+Press&rft.date=1984&rft.isbn=0-674-37935-7&rft.au=Bernard+Frischer&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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">Eduard Heine, <a href="//doi.org/10.1515/crll.1872.74.172" class="extiw" title="doi:10.1515/crll.1872.74.172">"Die Elemente der Functionenlehre"</a>, <i>[Crelle's] Journal für die reine und angewandte Mathematik</i>, No. 74 (1872): 172–188.</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">Georg Cantor, <a href="//doi.org/10.1007/BF01446819" class="extiw" title="doi:10.1007/BF01446819">"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5</a>, <i>Mathematische Annalen</i>, 21, 4 (1883‑12): 545–591.</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">Richard Dedekind, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=n-43AAAAMAAJ">Stetigkeit & irrationale Zahlen</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210709184745/https://books.google.ca/books?id=n-43AAAAMAAJ">Archived</a> 2021-07-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i> (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: <i>———, Gesammelte mathematische Werke</i>, ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334.</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">L. Kronecker, <a href="//doi.org/10.1515/crll.1887.101.337" class="extiw" title="doi:10.1515/crll.1887.101.337">"Ueber den Zahlbegriff"</a>, <i>[Crelle's] Journal für die reine und angewandte Mathematik</i>, No. 101 (1887): 337–355.</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">Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis", <i>Acta Academiae Scientiarum Imperialis Petropolitanae</i>, 1779, 1 (1779): 162–187.</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">Ramus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in: <i>Det Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger</i> (Kjoebenhavn: 1855), p. 106.</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">Eduard Heine, <a href="//doi.org/10.1515/crll.1859.56.87" class="extiw" title="doi:10.1515/crll.1859.56.87">"Einige Eigenschaften der <i>Lamé</i>schen Funktionen"</a>, <i>[Crelle's] Journal für die reine und angewandte Mathematik</i>, No. 56 (Jan. 1859): 87–99 at 97.</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">Siegmund Günther, <i>Darstellung der Näherungswerthe von Kettenbrüchen in independenter Form</i> (Erlangen: Eduard Besold, 1873); ———, "Kettenbruchdeterminanten", in: <i>Lehrbuch der Determinanten-Theorie: Für Studirende</i> (Erlangen: Eduard Besold, 1875), c. 6, pp. 156–186.</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:r1238218222"><cite id="CITEREFBogomolny" class="citation web cs1"><a href="/wiki/Cut-the-Knot" class="mw-redirect" title="Cut-the-Knot">Bogomolny, A.</a> <a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/numbers.shtml">"What's a number?"</a>. <i>Interactive Mathematics Miscellany and Puzzles</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100923231547/http://www.cut-the-knot.org/do_you_know/numbers.shtml">Archived</a> from the original on 23 September 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">11 July</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Interactive+Mathematics+Miscellany+and+Puzzles&rft.atitle=What%27s+a+number%3F&rft.aulast=Bogomolny&rft.aufirst=A.&rft_id=http%3A%2F%2Fwww.cut-the-knot.org%2Fdo_you_know%2Fnumbers.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFMartínez2007" class="citation journal cs1">Martínez, Alberto A. (2007). <a rel="nofollow" class="external text" href="https://www.martinezwritings.com/m/Euler_files/EulerMonthly.pdf">"Euler's 'mistake'? The radical product rule in historical perspective"</a> <span class="cs1-format">(PDF)</span>. <i>The American Mathematical Monthly</i>. <b>114</b> (4): 273–285. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2007.11920416">10.1080/00029890.2007.11920416</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:43778192">43778192</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Euler%27s+%27mistake%27%3F+The+radical+product+rule+in+historical+perspective&rft.volume=114&rft.issue=4&rft.pages=273-285&rft.date=2007&rft_id=info%3Adoi%2F10.1080%2F00029890.2007.11920416&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43778192%23id-name%3DS2CID&rft.aulast=Mart%C3%ADnez&rft.aufirst=Alberto+A.&rft_id=https%3A%2F%2Fwww.martinezwritings.com%2Fm%2FEuler_files%2FEulerMonthly.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"> <span class="citation mathworld" id="Reference-Mathworld-Natural_Number"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/NaturalNumber.html">"Natural Number"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Natural+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FNaturalNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span></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:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/natural%20number">"natural number"</a>. <i>Merriam-Webster.com</i>. <a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number">Archived</a> from the original on 13 December 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">4 October</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster.com&rft.atitle=natural+number&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fnatural%2520number&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuppes1972" class="citation book cs1"><a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick</a> (1972). <a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0/page/1"><i>Axiomatic Set Theory</i></a>. Courier Dover Publications. p. <a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0/page/1">1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61630-4" title="Special:BookSources/0-486-61630-4"><bdi>0-486-61630-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Set+Theory&rft.pages=1&rft.pub=Courier+Dover+Publications&rft.date=1972&rft.isbn=0-486-61630-4&rft.aulast=Suppes&rft.aufirst=Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faxiomaticsettheo00supp_0%2Fpage%2F1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"><span class="citation mathworld" id="Reference-Mathworld-Integer"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Integer.html">"Integer"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Integer&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FInteger.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span></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:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RepeatingDecimal.html">"Repeating Decimal"</a>. <i>Wolfram MathWorld</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200805170548/https://mathworld.wolfram.com/RepeatingDecimal.html">Archived</a> from the original on 5 August 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">23 July</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Repeating+Decimal&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRepeatingDecimal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-:1-38"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_38-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_38-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKontsevichZagier2001" class="citation cs2">Kontsevich, Maxim; Zagier, Don (2001), Engquist, Björn; Schmid, Wilfried (eds.), <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-642-56478-9_39">"Periods"</a>, <i>Mathematics Unlimited — 2001 and Beyond</i>, Berlin, Heidelberg: Springer, pp. 771–808, <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%2F978-3-642-56478-9_39">10.1007/978-3-642-56478-9_39</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-56478-9" title="Special:BookSources/978-3-642-56478-9"><bdi>978-3-642-56478-9</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">22 September</span> 2024</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Unlimited+%E2%80%94+2001+and+Beyond&rft.atitle=Periods&rft.pages=771-808&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F978-3-642-56478-9_39&rft.isbn=978-3-642-56478-9&rft.aulast=Kontsevich&rft.aufirst=Maxim&rft.au=Zagier%2C+Don&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-3-642-56478-9_39&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AlgebraicPeriod.html">"Algebraic Period"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">22 September</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=mathworld.wolfram.com&rft.atitle=Algebraic+Period&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAlgebraicPeriod.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></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:r1238218222"><cite id="CITEREFLagarias2013" class="citation journal cs1">Lagarias, Jeffrey C. (19 July 2013). "Euler's constant: Euler's work and modern developments". <i>Bulletin of the American Mathematical Society</i>. <b>50</b> (4): 527–628. <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/1303.1856">1303.1856</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/10.1090%2FS0273-0979-2013-01423-X">10.1090/S0273-0979-2013-01423-X</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0273-0979">0273-0979</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Euler%27s+constant%3A+Euler%27s+work+and+modern+developments&rft.volume=50&rft.issue=4&rft.pages=527-628&rft.date=2013-07-19&rft_id=info%3Aarxiv%2F1303.1856&rft.issn=0273-0979&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-2013-01423-X&rft.aulast=Lagarias&rft.aufirst=Jeffrey+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> <li id="cite_note-Saniga-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-Saniga_41-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanigaHolweckPracna2015" class="citation journal cs1">Saniga, Metod; Holweck, Frédéric; Pracna, Petr (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath3041192">"From Cayley-Dickson Algebras to Combinatorial Grassmannians"</a>. <i>Mathematics</i>. <b>3</b> (4). MDPI AG: 1192–1221. <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/1405.6888">1405.6888</a></span>. <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/10.3390%2Fmath3041192">10.3390/math3041192</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2227-7390">2227-7390</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics&rft.atitle=From+Cayley-Dickson+Algebras+to+Combinatorial+Grassmannians&rft.volume=3&rft.issue=4&rft.pages=1192-1221&rft.date=2015&rft_id=info%3Aarxiv%2F1405.6888&rft.issn=2227-7390&rft_id=info%3Adoi%2F10.3390%2Fmath3041192&rft.aulast=Saniga&rft.aufirst=Metod&rft.au=Holweck%2C+Fr%C3%A9d%C3%A9ric&rft.au=Pracna%2C+Petr&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fmath3041192&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><a href="/wiki/Tobias_Dantzig" title="Tobias Dantzig">Tobias Dantzig</a>, <i>Number, the language of science; a critical survey written for the cultured non-mathematician</i>, New York, The Macmillan Company, 1930.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Please supply an ISBN for this book.">ISBN missing</span></a></i>]</sup></li> <li>Erich Friedman, <i><a rel="nofollow" class="external text" href="https://www.stetson.edu/~efriedma/numbers.html">What's special about this number?</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180223062027/http://www2.stetson.edu/~efriedma/numbers.html">Archived</a> 2018-02-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i></li> <li>Steven Galovich, <i>Introduction to Mathematical Structures</i>, Harcourt Brace Javanovich, 1989, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-15-543468-3" title="Special:BookSources/0-15-543468-3">0-15-543468-3</a>.</li> <li><a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>, <i>Naive Set Theory</i>, Springer, 1974, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90092-6" title="Special:BookSources/0-387-90092-6">0-387-90092-6</a>.</li> <li><a href="/wiki/Morris_Kline" title="Morris Kline">Morris Kline</a>, <i>Mathematical Thought from Ancient to Modern Times</i>, Oxford University Press, 1990. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0195061352" title="Special:BookSources/978-0195061352">978-0195061352</a></li> <li><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> to *56, Cambridge University Press, 1910.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Please supply an ISBN for this book.">ISBN missing</span></a></i>]</sup></li> <li>Leo Cory, <i>A Brief History of Numbers</i>, Oxford University Press, 2015, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-870259-7" title="Special:BookSources/978-0-19-870259-7">978-0-19-870259-7</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Numbers" class="extiw" title="commons:Numbers"><span style="font-style:italic; font-weight:bold;">Numbers</span></a>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Number" class="extiw" title="q:Special:Search/Number">Number</a></b></i>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/number" class="extiw" title="wiktionary:number">number</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Primary_mathematics:_Numbers" class="extiw" title="v:Primary mathematics: Numbers">Primary mathematics: Numbers</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNechaev2001" class="citation cs1">Nechaev, V.I. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Number&oldid=11869">"Number"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Nechaev&rft.aufirst=V.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNumber%26oldid%3D11869&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTallant" class="citation web cs1">Tallant, Jonathan. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160308015528/http://www.numberphile.com/videos/exist.html">"Do Numbers Exist"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/exist.html">the original</a> on 8 March 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">6 April</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=Do+Numbers+Exist&rft.aulast=Tallant&rft.aufirst=Jonathan&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2Fexist.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220531120903/https://www.bbc.co.uk/programmes/p003hyd9"><i>In Our Time: Negative Numbers</i></a>. BBC Radio 4. 9 March 2006. Archived from <a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p003hyd9">the original</a> on 31 May 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=In+Our+Time%3A+Negative+Numbers&rft.pub=BBC+Radio+4&rft.date=2006-03-09&rft_id=http%3A%2F%2Fwww.bbc.co.uk%2Fprogrammes%2Fp003hyd9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobin_Wilson2007" class="citation web cs1">Robin Wilson (7 November 2007). <a rel="nofollow" class="external text" href="http://www.gresham.ac.uk/lectures-and-events/4000-years-of-numbers">"4000 Years of Numbers"</a>. <a href="/wiki/Gresham_College" title="Gresham College">Gresham College</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220408112133/http://www.gresham.ac.uk/lectures-and-events/4000-years-of-numbers">Archived</a> from the original on 8 April 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=4000+Years+of+Numbers&rft.pub=Gresham+College&rft.date=2007-11-07&rft.au=Robin+Wilson&rft_id=http%3A%2F%2Fwww.gresham.ac.uk%2Flectures-and-events%2F4000-years-of-numbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrulwich2011" class="citation news cs1">Krulwich, Robert (22 July 2011). <a rel="nofollow" class="external text" href="https://www.npr.org/sections/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1">"What's the World's Favorite Number?"</a>. <i>NPR</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210518141211/https://www.npr.org/sections/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1">Archived</a> from the original on 18 May 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NPR&rft.atitle=What%27s+the+World%27s+Favorite+Number%3F&rft.date=2011-07-22&rft.aulast=Krulwich&rft.aufirst=Robert&rft_id=https%3A%2F%2Fwww.npr.org%2Fsections%2Fkrulwich%2F2011%2F07%2F22%2F138493147%2Fwhat-s-your-favorite-number-world-wide-survey-v1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.npr.org/templates/transcript/transcript.php?storyId=139797360">"Cuddling With 9, Smooching With 8, Winking At 7"</a>. <i><a href="/wiki/NPR" title="NPR">NPR</a></i>. 21 August 2011. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181106205912/https://www.npr.org/templates/transcript/transcript.php?storyId=139797360?storyId=139797360">Archived</a> from the original on 6 November 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NPR&rft.atitle=Cuddling+With+9%2C+Smooching+With+8%2C+Winking+At+7&rft.date=2011-08-21&rft_id=https%3A%2F%2Fwww.npr.org%2Ftemplates%2Ftranscript%2Ftranscript.php%3FstoryId%3D139797360&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumber" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://oeis.org">Online Encyclopedia of Integer Sequences</a></li></ul> <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="Number_systems" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse 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:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> (<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>)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<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>)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<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>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<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 {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" 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 {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<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 {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<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>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<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 {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <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>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<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 {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" 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 {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a class="mw-selflink-fragment" href="#Main_classification">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></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"></div><div role="navigation" class="navbox" aria-labelledby="Number_theory" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#ffb;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_theory" title="Template:Number theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_theory" title="Template talk:Number theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_theory" title="Special:EditPage/Template:Number theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> (<a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>, <a href="/wiki/Non-abelian_class_field_theory" title="Non-abelian class field theory">non-abelian class field theory</a>, <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>, <a href="/wiki/Iwasawa%E2%80%93Tate_theory" class="mw-redirect" title="Iwasawa–Tate theory">Iwasawa–Tate theory</a>, <a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a>)</li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a> (<a href="/wiki/L-function" title="L-function">analytic theory of L-functions</a>, <a href="/wiki/Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a>)</li> <li><a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">Geometric number theory</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a> (<a href="/wiki/Arakelov_theory" title="Arakelov theory">Arakelov theory</a>, <a href="/wiki/Hodge%E2%80%93Arakelov_theory" title="Hodge–Arakelov theory">Hodge–Arakelov theory</a>)</li> <li><a href="/wiki/Arithmetic_combinatorics" title="Arithmetic combinatorics">Arithmetic combinatorics</a> (<a href="/wiki/Additive_number_theory" title="Additive number theory">additive number theory</a>)</li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic geometry</a> (<a href="/wiki/Anabelian_geometry" title="Anabelian geometry">anabelian geometry</a>, <a href="/wiki/P-adic_Hodge_theory" title="P-adic Hodge theory">P-adic Hodge theory</a>)</li> <li><a href="/wiki/Arithmetic_topology" title="Arithmetic topology">Arithmetic topology</a></li> <li><a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Numbers</a></li> <li><a href="/wiki/0" title="0">0</a></li> <li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></li> <li><a href="/wiki/1" title="1">Unity</a></li> <li><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number">P-adic numbers</a> (<a href="/wiki/P-adic_analysis" title="P-adic analysis">P-adic analysis</a>)</li> <li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></li> <li><a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li> <li><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Advanced concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quadratic_form" title="Quadratic form">Quadratic forms</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular forms</a></li> <li><a href="/wiki/L-function" title="L-function">L-functions</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a></li> <li><a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a></li> <li><a href="/wiki/Irrationality_measure" title="Irrationality measure">Irrationality measure</a></li> <li><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><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> <a href="/wiki/Category:Number_theory" title="Category:Number theory">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_number_theory_topics" title="List of number theory topics">List of topics</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_recreational_number_theory_topics" title="List of recreational number theory topics">List of recreational topics</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo.svg" class="mw-file-description" title="Wikibooks page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> <a href="https://en.wikibooks.org/wiki/Number_Theory" class="extiw" title="wikibooks:Number Theory">Wikibook</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://en.wikiversity.org/wiki/Number_Theory" class="extiw" title="wikiversity:Number Theory">Wikiversity</a></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></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11563#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></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/4067271-2">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85093206">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119326327">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119326327">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00571509">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="čísla"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph202644&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX4678887">Spain</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538634905171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Number&oldid=1256779505">https://en.wikipedia.org/w/index.php?title=Number&oldid=1256779505</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">Categories</a>: <ul><li><a href="/wiki/Category:Numbers" title="Category:Numbers">Numbers</a></li><li><a href="/wiki/Category:Group_theory" title="Category:Group theory">Group theory</a></li><li><a href="/wiki/Category:Abstraction" title="Category:Abstraction">Abstraction</a></li><li><a href="/wiki/Category:Mathematical_objects" title="Category:Mathematical objects">Mathematical objects</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:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><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:Wikipedia_indefinitely_semi-protected_pages" title="Category:Wikipedia indefinitely semi-protected pages">Wikipedia indefinitely semi-protected pages</a></li><li><a href="/wiki/Category:Wikipedia_indefinitely_move-protected_pages" title="Category:Wikipedia indefinitely move-protected pages">Wikipedia indefinitely move-protected pages</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_December_2022" title="Category:Use dmy dates from December 2022">Use dmy dates from December 2022</a></li><li><a href="/wiki/Category:All_articles_lacking_reliable_references" title="Category:All articles lacking reliable references">All articles lacking reliable references</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_January_2017" title="Category:Articles lacking reliable references from January 2017">Articles lacking reliable references from January 2017</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_November_2022" title="Category:Articles needing additional references from November 2022">Articles needing additional references from November 2022</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_containing_Sanskrit-language_text" title="Category:Articles containing Sanskrit-language text">Articles containing Sanskrit-language text</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_September_2020" title="Category:Articles lacking reliable references from September 2020">Articles lacking reliable references from September 2020</a></li><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2020" title="Category:Articles with unsourced statements from September 2020">Articles with unsourced statements from September 2020</a></li><li><a href="/wiki/Category:Articles_containing_German-language_text" title="Category:Articles containing German-language text">Articles containing German-language text</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_September_2020" title="Category:Wikipedia articles needing clarification from September 2020">Wikipedia articles needing clarification from September 2020</a></li><li><a href="/wiki/Category:Pages_with_missing_ISBNs" title="Category:Pages with missing ISBNs">Pages with missing ISBNs</a></li><li><a href="/wiki/Category:Articles_prone_to_spam_from_July_2022" title="Category:Articles prone to spam from July 2022">Articles prone to spam from July 2022</a></li><li><a href="/wiki/Category:Commons_link_is_on_Wikidata" title="Category:Commons link is on Wikidata">Commons link is on Wikidata</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 11 November 2024, at 14:51<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=Number&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"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></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"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-khxd6","wgBackendResponseTime":169,"wgPageParseReport":{"limitreport":{"cputime":"1.515","walltime":"1.938","ppvisitednodes":{"value":8915,"limit":1000000},"postexpandincludesize":{"value":164671,"limit":2097152},"templateargumentsize":{"value":11280,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":34,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":191533,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1588.706 1 -total"," 23.68% 376.219 2 Template:Reflist"," 10.63% 168.905 7 Template:Lang"," 8.47% 134.615 8 Template:Fix"," 7.70% 122.379 6 Template:Cite_journal"," 7.57% 120.267 2 Template:Navbox"," 7.28% 115.694 1 Template:Number_systems"," 6.69% 106.251 4 Template:Better_source_needed"," 5.66% 89.940 1 Template:Short_description"," 5.01% 79.607 14 Template:Cite_book"]},"scribunto":{"limitreport-timeusage":{"value":"0.916","limit":"10.000"},"limitreport-memusage":{"value":22141411,"limit":52428800}},"cachereport":{"origin":"mw-api-ext.codfw.main-7556f8b5dd-lp595","timestamp":"20241122140451","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Number","url":"https:\/\/en.wikipedia.org\/wiki\/Number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q11563","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q11563","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":"2001-09-27T19:05:04Z","dateModified":"2024-11-11T14:51:40Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a0\/NumberSetinC.svg","headline":"mathematical object used to count, label, and measure"}</script> </body> </html>