Fraction - Wikipedia

<!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>Fraction - 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":"084f55b1-a31b-4702-9d62-13b6631c7eb5","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Fraction","wgTitle":"Fraction","wgCurRevisionId":1258329595,"wgRevisionId":1258329595,"wgArticleId":1704824,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description is different from Wikidata","Wikipedia indefinitely semi-protected pages","Articles containing Latin-language text","Articles needing additional references from June 2023","All articles needing additional references","All articles with unsourced statements","Articles with unsourced statements from February 2016","Articles containing Sanskrit-language text","Commons category link is on Wikidata", "Fractions (mathematics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Fraction","wgRelevantArticleId":1704824,"wgIsProbablyEditable":false,"wgRelevantPageIsProbablyEditable":false,"wgRestrictionEdit":["autoconfirmed"],"wgRestrictionMove":["autoconfirmed"],"wgRedirectedFrom":"Fraction_(mathematics)","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,"wgInternalRedirectTargetUrl":"/wiki/Fraction","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition": "interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q66055","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.math.styles":"ready","ext.cite.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=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","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/b/b9/Cake_quarters.svg/1200px-Cake_quarters.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="912"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/800px-Cake_quarters.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="608"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/640px-Cake_quarters.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="486"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Fraction - 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/Fraction"> <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/Fraction"> <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-Fraction rootpage-Fraction 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=Fraction" 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=Fraction" 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=Fraction" 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=Fraction" 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-Vocabulary" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vocabulary"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Vocabulary</span> </div> </a> <ul id="toc-Vocabulary-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forms_of_fractions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Forms_of_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Forms of fractions</span> </div> </a> <button aria-controls="toc-Forms_of_fractions-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 Forms of fractions subsection</span> </button> <ul id="toc-Forms_of_fractions-sublist" class="vector-toc-list"> <li id="toc-Simple,_common,_or_vulgar_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple,_common,_or_vulgar_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Simple, common, or vulgar fractions</span> </div> </a> <ul id="toc-Simple,_common,_or_vulgar_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proper_and_improper_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proper_and_improper_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Proper and improper fractions</span> </div> </a> <ul id="toc-Proper_and_improper_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocals_and_the_"invisible_denominator"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reciprocals_and_the_"invisible_denominator""> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Reciprocals and the "invisible denominator"</span> </div> </a> <ul id="toc-Reciprocals_and_the_"invisible_denominator"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ratios" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ratios"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Ratios</span> </div> </a> <ul id="toc-Ratios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decimal_fractions_and_percentages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decimal_fractions_and_percentages"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Decimal fractions and percentages</span> </div> </a> <ul id="toc-Decimal_fractions_and_percentages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixed_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixed_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Mixed numbers</span> </div> </a> <ul id="toc-Mixed_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_notions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Historical notions</span> </div> </a> <ul id="toc-Historical_notions-sublist" class="vector-toc-list"> <li id="toc-Egyptian_fraction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Egyptian_fraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.1</span> <span>Egyptian fraction</span> </div> </a> <ul id="toc-Egyptian_fraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_and_compound_fractions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complex_and_compound_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.2</span> <span>Complex and compound fractions</span> </div> </a> <ul id="toc-Complex_and_compound_fractions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Arithmetic_with_fractions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Arithmetic_with_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Arithmetic with fractions</span> </div> </a> <button aria-controls="toc-Arithmetic_with_fractions-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 Arithmetic with fractions subsection</span> </button> <ul id="toc-Arithmetic_with_fractions-sublist" class="vector-toc-list"> <li id="toc-Equivalent_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Equivalent fractions</span> </div> </a> <ul id="toc-Equivalent_fractions-sublist" class="vector-toc-list"> <li id="toc-Simplifying_(reducing)_fractions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Simplifying_(reducing)_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Simplifying (reducing) fractions</span> </div> </a> <ul id="toc-Simplifying_(reducing)_fractions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparing_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparing_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Comparing fractions</span> </div> </a> <ul id="toc-Comparing_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Addition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Addition</span> </div> </a> <ul id="toc-Addition-sublist" class="vector-toc-list"> <li id="toc-Adding_unlike_quantities" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Adding_unlike_quantities"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Adding unlike quantities</span> </div> </a> <ul id="toc-Adding_unlike_quantities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subtraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Subtraction</span> </div> </a> <ul id="toc-Subtraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplication"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Multiplication</span> </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> <li id="toc-Multiplying_a_fraction_by_another_fraction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiplying_a_fraction_by_another_fraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.1</span> <span>Multiplying a fraction by another fraction</span> </div> </a> <ul id="toc-Multiplying_a_fraction_by_another_fraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplying_a_fraction_by_a_whole_number" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiplying_a_fraction_by_a_whole_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.2</span> <span>Multiplying a fraction by a whole number</span> </div> </a> <ul id="toc-Multiplying_a_fraction_by_a_whole_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplying_mixed_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiplying_mixed_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.3</span> <span>Multiplying mixed numbers</span> </div> </a> <ul id="toc-Multiplying_mixed_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Division</span> </div> </a> <ul id="toc-Division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Converting_between_decimals_and_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Converting_between_decimals_and_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Converting between decimals and fractions</span> </div> </a> <ul id="toc-Converting_between_decimals_and_fractions-sublist" class="vector-toc-list"> <li id="toc-Converting_repeating_decimals_to_fractions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Converting_repeating_decimals_to_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7.1</span> <span>Converting repeating decimals to fractions</span> </div> </a> <ul id="toc-Converting_repeating_decimals_to_fractions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Fractions_in_abstract_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fractions_in_abstract_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Fractions in abstract mathematics</span> </div> </a> <ul id="toc-Fractions_in_abstract_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_fractions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algebraic_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Algebraic fractions</span> </div> </a> <ul id="toc-Algebraic_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radical_expressions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Radical_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Radical expressions</span> </div> </a> <ul id="toc-Radical_expressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Typographical_variations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Typographical_variations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Typographical variations</span> </div> </a> <ul id="toc-Typographical_variations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_formal_education" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_formal_education"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>In formal education</span> </div> </a> <button aria-controls="toc-In_formal_education-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In formal education subsection</span> </button> <ul id="toc-In_formal_education-sublist" class="vector-toc-list"> <li id="toc-Primary_schools" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primary_schools"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Primary schools</span> </div> </a> <ul id="toc-Primary_schools-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Documents_for_teachers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Documents_for_teachers"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Documents for teachers</span> </div> </a> <ul id="toc-Documents_for_teachers-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">10</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">11</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">12</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">13</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">Fraction</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 99 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-99" 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">99 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/Breuk_(wiskunde)" title="Breuk (wiskunde) – Afrikaans" lang="af" hreflang="af" data-title="Breuk (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B3%D8%B1_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="كسر (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="كسر (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AD%E0%A6%97%E0%A7%8D%E0%A6%A8%E0%A6%BE%E0%A6%82%E0%A6%B6_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="ভগ্নাংশ (গণিত) – Assamese" lang="as" hreflang="as" data-title="ভগ্নাংশ (গণিত)" 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/Fraici%C3%B3n" title="Fraición – Asturian" lang="ast" hreflang="ast" data-title="Fraición" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Pachjta" title="Pachjta – Aymara" lang="ay" hreflang="ay" data-title="Pachjta" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/K%C9%99sr" title="Kəsr – Azerbaijani" lang="az" hreflang="az" data-title="Kəsr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A6%97%E0%A7%8D%E0%A6%A8%E0%A6%BE%E0%A6%82%E0%A6%B6" title="ভগ্নাংশ – Bangla" lang="bn" hreflang="bn" data-title="ভগ্নাংশ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D3%99%D1%81%D0%B5%D1%80%D2%99%D3%99%D1%80" title="Кәсерҙәр – Bashkir" lang="ba" hreflang="ba" data-title="Кәсерҙәр" 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%94%D1%80%D0%BE%D0%B1" title="Дроб – Belarusian" lang="be" hreflang="be" data-title="Дроб" 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%94%D1%80%D0%BE%D0%B1%D1%8B" title="Дробы – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дробы" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Praksyon" title="Praksyon – Central Bikol" lang="bcl" hreflang="bcl" data-title="Praksyon" 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%94%D1%80%D0%BE%D0%B1_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Дроб (математика) – Bulgarian" lang="bg" hreflang="bg" data-title="Дроб (математика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Rann" title="Rann – Breton" lang="br" hreflang="br" data-title="Rann" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fracci%C3%B3" title="Fracció – Catalan" lang="ca" hreflang="ca" data-title="Fracció" 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%92%D0%B0%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Вак (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Вак (математика)" 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/Zlomek" title="Zlomek – Czech" lang="cs" hreflang="cs" data-title="Zlomek" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Rupande" title="Rupande – Shona" lang="sn" hreflang="sn" data-title="Rupande" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffracsiwn" title="Ffracsiwn – Welsh" lang="cy" hreflang="cy" data-title="Ffracsiwn" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Br%C3%B8k" title="Brøk – Danish" lang="da" hreflang="da" data-title="Brøk" 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/Bruchrechnung" title="Bruchrechnung – German" lang="de" hreflang="de" data-title="Bruchrechnung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Murdarv" title="Murdarv – Estonian" lang="et" hreflang="et" data-title="Murdarv" 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%9A%CE%BB%CE%AC%CF%83%CE%BC%CE%B1" title="Κλάσμα – Greek" lang="el" hreflang="el" data-title="Κλάσμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fracci%C3%B3n" title="Fracción – Spanish" lang="es" hreflang="es" data-title="Fracción" 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/Frakcio_(matematiko)" title="Frakcio (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Frakcio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zatiki_(matematika)" title="Zatiki (matematika) – Basque" lang="eu" hreflang="eu" data-title="Zatiki (matematika)" 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/%DA%A9%D8%B3%D8%B1" title="کسر – Persian" lang="fa" hreflang="fa" data-title="کسر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fraction_(math%C3%A9matiques)" title="Fraction (mathématiques) – French" lang="fr" hreflang="fr" data-title="Fraction (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Bloigh_(matamataig)" title="Bloigh (matamataig) – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Bloigh (matamataig)" 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/Fracci%C3%B3n_(matem%C3%A1ticas)" title="Fracción (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Fracción (matemáticas)" 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/%E5%88%86%E6%95%B8" title="分數 – Gan" lang="gan" hreflang="gan" data-title="分數" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EC%88%98_(%EC%88%98%ED%95%99)" title="분수 (수학) – Korean" lang="ko" hreflang="ko" data-title="분수 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%BF%D5%B8%D6%80%D5%A1%D5%AF_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Կոտորակ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Կոտորակ (մաթեմատիկա)" 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%AD%E0%A4%BF%E0%A4%A8%E0%A5%8D%E0%A4%A8" title="भिन्न – Hindi" lang="hi" hreflang="hi" data-title="भिन्न" 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/Razlomak" title="Razlomak – Croatian" lang="hr" hreflang="hr" data-title="Razlomak" 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/Fraciono_(matematiko)" title="Fraciono (matematiko) – Ido" lang="io" hreflang="io" data-title="Fraciono (matematiko)" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pecahan" title="Pecahan – Indonesian" lang="id" hreflang="id" data-title="Pecahan" 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-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Fraction_(yemathematika)" title="I-Fraction (yemathematika) – Xhosa" lang="xh" hreflang="xh" data-title="I-Fraction (yemathematika)" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Almennt_brot" title="Almennt brot – Icelandic" lang="is" hreflang="is" data-title="Almennt brot" 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/Frazione_(matematica)" title="Frazione (matematica) – Italian" lang="it" hreflang="it" data-title="Frazione (matematica)" 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%A9%D7%91%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="שבר (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="שבר (מתמטיקה)" 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_pecahan" title="Wilangan pecahan – Javanese" lang="jv" hreflang="jv" data-title="Wilangan pecahan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AD%E0%B2%BF%E0%B2%A8%E0%B3%8D%E0%B2%A8%E0%B2%BE%E0%B2%82%E0%B2%95" title="ಭಿನ್ನಾಂಕ – Kannada" lang="kn" hreflang="kn" data-title="ಭಿನ್ನಾಂಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Fraksyon_(mat%C3%A9matik)" title="Fraksyon (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Fraksyon (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D1%87%D3%A9%D0%BA_(%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0)" title="Бөлчөк (арифметика) – Kyrgyz" lang="ky" hreflang="ky" data-title="Бөлчөк (арифметика)" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Fractio_(mathematica)" title="Fractio (mathematica) – Latin" lang="la" hreflang="la" data-title="Fractio (mathematica)" 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/Da%C4%BCskaitlis" title="Daļskaitlis – Latvian" lang="lv" hreflang="lv" data-title="Daļskaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trupmena" title="Trupmena – Lithuanian" lang="lt" hreflang="lt" data-title="Trupmena" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Frato" title="Frato – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Frato" 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-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Omukutule_ogwa_bulijjo(Common_fraction)" title="Omukutule ogwa bulijjo(Common fraction) – Ganda" lang="lg" hreflang="lg" data-title="Omukutule ogwa bulijjo(Common fraction)" 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/Frazzion" title="Frazzion – Lombard" lang="lmo" hreflang="lmo" data-title="Frazzion" 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%A1mol%C3%A1s_t%C3%B6rtekkel" title="Számolás törtekkel – Hungarian" lang="hu" hreflang="hu" data-title="Számolás törtekkel" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D1%80%D0%BE%D0%BF%D0%BA%D0%B0" title="Дропка – Macedonian" lang="mk" hreflang="mk" data-title="Дропка" 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/Ampaha_(matematika)" title="Ampaha (matematika) – Malagasy" lang="mg" hreflang="mg" data-title="Ampaha (matematika)" 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%AD%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%A8%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="ഭിന്നസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="ഭിന്നസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Frazzjoni_(matematika)" title="Frazzjoni (matematika) – Maltese" lang="mt" hreflang="mt" data-title="Frazzjoni (matematika)" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pecahan" title="Pecahan – Malay" lang="ms" hreflang="ms" data-title="Pecahan" 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-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%95%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="အပိုင်းကိန်း – Burmese" lang="my" hreflang="my" data-title="အပိုင်းကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Breuk_(wiskunde)" title="Breuk (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Breuk (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0" title="分数 – Japanese" lang="ja" hreflang="ja" data-title="分数" 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/Br%C3%B6%C3%B6ktaal" title="Brööktaal – Northern Frisian" lang="frr" hreflang="frr" data-title="Brööktaal" 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/Br%C3%B8k" title="Brøk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Brøk" 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/Br%C3%B8k" title="Brøk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Brøk" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Fraccion_(matematicas)" title="Fraccion (matematicas) – Occitan" lang="oc" hreflang="oc" data-title="Fraccion (matematicas)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Hirmeemmii_(fraction)" title="Hirmeemmii (fraction) – Oromo" lang="om" hreflang="om" data-title="Hirmeemmii (fraction)" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://uz.wikipedia.org/wiki/Kasr" title="Kasr – Uzbek" lang="uz" hreflang="uz" data-title="Kasr" 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-blk mw-list-item"><a href="https://blk.wikipedia.org/wiki/%E1%80%82%E1%80%8F%E1%80%94%E1%80%BA%EA%A9%BB%E1%80%81%E1%80%B1%E1%80%AB%E1%80%9D%E1%80%BA" title="ဂဏန်ꩻခေါဝ် – Pa'O" lang="blk" hreflang="blk" data-title="ဂဏန်ꩻခေါဝ်" data-language-autonym="ပအိုဝ်ႏဘာႏသာႏ" data-language-local-name="Pa'O" class="interlanguage-link-target"><span>ပအိုဝ်ႏဘာႏသာႏ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Frakshan_(matimatix)" title="Frakshan (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Frakshan (matimatix)" 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/Br%C3%B6%C3%B6k" title="Bröök – Low German" lang="nds" hreflang="nds" data-title="Bröök" 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/U%C5%82amek" title="Ułamek – Polish" lang="pl" hreflang="pl" data-title="Ułamek" 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/Fra%C3%A7%C3%A3o" title="Fração – Portuguese" lang="pt" hreflang="pt" data-title="Fração" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Frac%C8%9Bie" title="Fracție – Romanian" lang="ro" hreflang="ro" data-title="Fracție" 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/Ch%27iqtaku" title="Ch'iqtaku – Quechua" lang="qu" hreflang="qu" data-title="Ch'iqtaku" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D1%80%D0%BE%D0%B1%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Дробь (математика) – Russian" lang="ru" hreflang="ru" data-title="Дробь (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Thyesa" title="Thyesa – Albanian" lang="sq" hreflang="sq" data-title="Thyesa" 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/Frazzioni_(matim%C3%A0tica)" title="Frazzioni (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Frazzioni (matimàtica)" 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/Fraction_(mathematics)" title="Fraction (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fraction (mathematics)" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zlomok_(matematika)" title="Zlomok (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Zlomok (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/Ulomek" title="Ulomek – Slovenian" lang="sl" hreflang="sl" data-title="Ulomek" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%DB%95%D8%B1%D8%AA" title="کەرت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کەرت" 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%A0%D0%B0%D0%B7%D0%BB%D0%BE%D0%BC%D0%B0%D0%BA" title="Разломак – Serbian" lang="sr" hreflang="sr" data-title="Разломак" 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/Razlomak" title="Razlomak – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Razlomak" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Murtoluku" title="Murtoluku – Finnish" lang="fi" hreflang="fi" data-title="Murtoluku" 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/Br%C3%A5k" title="Bråk – Swedish" lang="sv" hreflang="sv" data-title="Bråk" 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/Hatimbilang" title="Hatimbilang – Tagalog" lang="tl" hreflang="tl" data-title="Hatimbilang" 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%AA%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AE%AE%E0%AF%8D" title="பின்னம் – Tamil" lang="ta" hreflang="ta" data-title="பின்னம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tir%E1%BA%93i_(tusnakt)" title="Tirẓi (tusnakt) – Kabyle" lang="kab" hreflang="kab" data-title="Tirẓi (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AD%E0%B0%BF%E0%B0%A8%E0%B1%8D%E0%B0%A8%E0%B0%82" title="భిన్నం – Telugu" lang="te" hreflang="te" data-title="భిన్నం" 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%B9%80%E0%B8%A8%E0%B8%A9%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99" title="เศษส่วน – Thai" lang="th" hreflang="th" data-title="เศษส่วน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kesir" title="Kesir – Turkish" lang="tr" hreflang="tr" data-title="Kesir" 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/Drob" title="Drob – Turkmen" lang="tk" hreflang="tk" data-title="Drob" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%80%D1%96%D0%B1" title="Дріб – Ukrainian" lang="uk" hreflang="uk" data-title="Дріб" 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/%DA%A9%D8%B3%D8%B1_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="کسر (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="کسر (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%83%DB%95%D8%B3%D9%89%D8%B1" title="كەسىر – Uyghur" lang="ug" hreflang="ug" data-title="كەسىر" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_s%E1%BB%91" title="Phân số – Vietnamese" lang="vi" hreflang="vi" data-title="Phân 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Fraksyon" title="Fraksyon – Waray" lang="war" hreflang="war" data-title="Fraksyon" 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/%E5%88%86%E6%95%B0" title="分数 – Wu" lang="wuu" hreflang="wuu" data-title="分数" 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-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%91%D7%A8%D7%90%D7%9B%D7%98%D7%99%D7%99%D7%9C" title="בראכטייל – Yiddish" lang="yi" hreflang="yi" data-title="בראכטייל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%88%86%E6%95%B8" title="分數 – Cantonese" lang="yue" hreflang="yue" data-title="分數" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%86%E6%95%B8" title="分數 – Chinese" lang="zh" hreflang="zh" data-title="分數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q66055#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/Fraction" 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:Fraction" 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/Fraction"><span>Read</span></a></li><li id="ca-viewsource" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Fraction&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=Fraction&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/Fraction"><span>Read</span></a></li><li id="ca-more-viewsource" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Fraction&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=Fraction&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/Fraction" 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/Fraction" 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=Fraction&oldid=1258329595" 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=Fraction&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=Fraction&id=1258329595&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%2FFraction"><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%2FFraction"><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=Fraction&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=Fraction&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/Category:Fractions" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/A_Guide_to_the_GRE/Fractions" hreflang="en"><span>Wikibooks</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/Q66055" 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"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Fraction_(mathematics)&redirect=no" class="mw-redirect" title="Fraction (mathematics)">Fraction (mathematics)</a>)</span></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">Ratio of two numbers</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/Fraction_(disambiguation)" class="mw-disambig" title="Fraction (disambiguation)">Fraction (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cake_quarters.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/220px-Cake_quarters.svg.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/330px-Cake_quarters.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/440px-Cake_quarters.svg.png 2x" data-file-width="504" data-file-height="383" /></a><figcaption>A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the <span class="nowrap">fraction <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">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></span></figcaption></figure> <p>A <b>fraction</b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la">fractus</i>, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A <i>common</i>, <i>vulgar</i>, or <i>simple</i> fraction (examples: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {17}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>17</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {17}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af3ddb2b7911535514c1999a286cfc7a4aa5272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.843ex;" alt="{\displaystyle {\tfrac {17}{3}}}"></span>) consists of an integer <b>numerator</b>, displayed above a line (or before a slash like <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), and a <a href="/wiki/Division_by_zero" title="Division by zero">non-zero</a> integer <b>denominator</b>, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> of a cake. </p><p>Fractions can be used to represent <a href="/wiki/Ratio" title="Ratio">ratios</a> and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Thus the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division <span class="nowrap">3 ÷ 4</span> (three divided by four). </p><p>We can also write negative fractions, which represent the opposite of a positive fraction. For example, if <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> represents a half-dollar profit, then −<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> represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −<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>, <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">1</span><span class="sr-only">/</span><span class="den">−2</span></span>⁠</span> all represent the same fraction –  negative one-half. And because a negative divided by a negative produces a positive, <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> represents positive one-half. </p><p>In mathematics a <a href="/wiki/Rational_number" title="Rational number">rational number</a> is a number that can be represented by a fraction of the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><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 zero; the set of all rational numbers is commonly represented by the symbol <b>Q</b> or <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>⁠</span>, which stands for <a href="/wiki/Quotient" title="Quotient">quotient</a>. The term <i>fraction</i> and the notation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><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> can also be used for mathematical expressions that do not represent a rational number (for example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed29ed3827e5df443031c6a1e01dabc3c7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.027ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}}"></span>), and even do not represent any number (for example the <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fraction</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 \textstyle {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9ca3230ba1cdf6e52ca4c58d03819937fcddc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.776ex; height:3.343ex;" alt="{\displaystyle \textstyle {\frac {1}{x}}}"></span>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Vocabulary">Vocabulary</h2></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/Numeral_(linguistics)#Fractional_numbers" title="Numeral (linguistics)">Numeral (linguistics) § Fractional numbers</a>, <a href="/wiki/English_numerals#Fractions_and_decimals" title="English numerals">English numerals § Fractions and decimals</a>, and <a href="/wiki/Unicode_subscripts_and_superscripts#Fraction_slash" title="Unicode subscripts and superscripts">Unicode subscripts and superscripts § Fraction slash</a></div> <p>In a fraction, the number of equal parts being described is the <b>numerator</b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la">numerātor</i>, "counter" or "numberer"), and the type or variety of the parts is the <b>denominator</b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la">dēnōminātor</i>, "thing that names or designates").<sup id="cite_ref-schwartzman_2-0" class="reference"><a href="#cite_note-schwartzman-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>3<span class="cite-bracket">]</span></a></sup> As an example, the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> amounts to eight parts, each of which is of the type named "fifth". In terms of <a href="/wiki/Division_(math)" class="mw-redirect" title="Division (math)">division</a>, the numerator corresponds to the <a href="/wiki/Division_(math)" class="mw-redirect" title="Division (math)">dividend</a>, and the denominator corresponds to the <a href="/wiki/Division_(math)" class="mw-redirect" title="Division (math)">divisor</a>. </p><p>Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a <b>fraction bar</b>. The fraction bar may be horizontal (as in <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>), oblique (as in 2/5), or diagonal (as in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">4</span>⁄<span class="den">9</span></span>).<sup id="cite_ref-ambrose_4-0" class="reference"><a href="#cite_note-ambrose-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These marks are respectively known as the horizontal bar; the virgule, <a href="/wiki/Slash_mark" class="mw-redirect" title="Slash mark">slash</a> (<a href="/wiki/American_English" title="American English">US</a>), or <a href="/wiki/Oblique_stroke" class="mw-redirect" title="Oblique stroke">stroke</a> (<a href="/wiki/British_English" title="British English">UK</a>); and the fraction bar, solidus,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Fraction_slash" class="mw-redirect" title="Fraction slash">fraction slash</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>n 1<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Typography" title="Typography">typography</a>, fractions stacked vertically are also known as "<a href="/wiki/En_(typography)" title="En (typography)">en</a>" or "<a href="/wiki/En_dash" class="mw-redirect" title="En dash">nut</a> fractions", and diagonal ones as "<a href="/wiki/Em_(typography)" title="Em (typography)">em</a>" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow <i>en</i> square, or a wider <i>em</i> square.<sup id="cite_ref-ambrose_4-1" class="reference"><a href="#cite_note-ambrose-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In traditional <a href="/wiki/Typefounding" class="mw-redirect" title="Typefounding">typefounding</a>, a piece of type bearing a complete fraction (e.g. <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>) was known as a "case fraction", while those representing only part of fraction were called "piece fractions". </p><p>The denominators of English fractions are generally expressed as <a href="/wiki/Ordinal_number_(linguistics)#Variations" class="mw-redirect" title="Ordinal number (linguistics)">ordinal numbers</a>, in the plural if the numerator is not 1. (For example, <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">5</span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> are both read as a number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "<a href="/wiki/Percent" class="mw-redirect" title="Percent">percent</a>". </p><p>When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read out as a whole number. For example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span> may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter"). </p><p>The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction <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">5</span></span>⁠</span> and "two fifths" is the same fraction understood as 2 instances of <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">5</span></span>⁠</span>.) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>. (For example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span> may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a <a href="/wiki/Slash_mark" class="mw-redirect" title="Slash mark">slash mark</a>. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are <i>not</i> powers of ten are often rendered in this fashion (e.g., <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">117</span></span>⁠</span> as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">6</span><span class="sr-only">/</span><span class="den">1000000</span></span>⁠</span> as "six-millionths", "six millionths", or "six one-millionths"). </p> <div class="mw-heading mw-heading2"><h2 id="Forms_of_fractions">Forms of fractions</h2></div> <div class="mw-heading mw-heading3"><h3 id="Simple,_common,_or_vulgar_fractions"><span id="Simple.2C_common.2C_or_vulgar_fractions"></span>Simple, common, or vulgar fractions<span class="anchor" id="Simple_fraction"></span><span class="anchor" id="Common_fraction"></span><span class="anchor" id="Vulgar_fraction"></span></h3></div> <p>A <b>simple fraction</b> (also known as a <b>common fraction</b> or <b>vulgar fraction</b>, where vulgar is Latin for "common") is a <a href="/wiki/Rational_number" title="Rational number">rational number</a> written as <i>a</i>/<i>b</i> or <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span>, where <i>a</i> and <i>b</i> are both <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> As with other fractions, the denominator (<i>b</i>) cannot be zero. Examples include <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>, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">−8</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">−5</span></span>⁠</span>. The term was originally used to distinguish this type of fraction from the <a href="/wiki/Sexagesimal#Fractions" title="Sexagesimal">sexagesimal fraction</a> used in astronomy.<sup id="cite_ref-Smith1958_11-0" class="reference"><a href="#cite_note-Smith1958-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not <i>common fractions</i>; though, unless irrational, they can be evaluated to a common fraction. </p> <ul><li>A <i><a href="/wiki/Unit_fraction" title="Unit fraction">unit fraction</a></i> is a common fraction with a numerator of 1 (e.g., <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">7</span></span>⁠</span>). Unit fractions can also be expressed using negative exponents, as in 2<sup>−1</sup>, which represents 1/2, and 2<sup>−2</sup>, which represents 1/(2<sup>2</sup>) or 1/4.</li> <li>A <i><a href="/wiki/Dyadic_rational" title="Dyadic rational">dyadic fraction</a></i> is a common fraction in which the denominator is a <a href="/wiki/Power_of_two" title="Power of two">power of two</a>, e.g. <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">8</span></span>⁠</span> = <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<sup>3</sup></span></span>⁠</span>.</li></ul> <p>In Unicode, precomposed fraction characters are in the <a href="/wiki/Number_Forms" title="Number Forms">Number Forms</a> block. </p> <div class="mw-heading mw-heading3"><h3 id="Proper_and_improper_fractions">Proper and improper fractions</h3></div> <p>Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.<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> The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a piece", so a proper fraction must be less than 1.<sup id="cite_ref-Smith1958_11-1" class="reference"><a href="#cite_note-Smith1958-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This was explained in the 17th century textbook <i><a href="/wiki/The_Ground_of_Arts" title="The Ground of Arts">The Ground of Arts</a></i>.<sup id="cite_ref-Williams2011_13-0" class="reference"><a href="#cite_note-Williams2011-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Record1654_14-0" class="reference"><a href="#cite_note-Record1654-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In general, a common fraction is said to be a <b>proper fraction</b>, if the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It is said to be an <b>improper fraction</b>, or sometimes <b>top-heavy fraction</b>,<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> if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3. </p> <div class="mw-heading mw-heading3"><h3 id="Reciprocals_and_the_"invisible_denominator""><span id="Reciprocals_and_the_.22invisible_denominator.22"></span>Reciprocals and the "invisible denominator"</h3></div> <p>The <i><a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocal</a></i> of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span>, for instance, is <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">3</span></span>⁠</span>. The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. </p><p>When the numerator and denominator of a fraction are equal (for example, <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">7</span></span>⁠</span>), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. </p><p>Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">17</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span>, where 1 is sometimes referred to as the <i>invisible denominator</i>.<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> Therefore, every fraction or integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is <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">17</span></span>⁠</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ratios">Ratios</h3></div> <p>A <i><a href="/wiki/Ratio" title="Ratio">ratio</a></i> is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group <i>n</i>". For example, if a car lot had 12 vehicles, of which </p> <ul><li>2 are white,</li> <li>6 are red, and</li> <li>4 are yellow,</li></ul> <p>then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1. </p><p>A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> of the cars or <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> of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or <a href="/wiki/Probability" title="Probability">probability</a> that it would be yellow. </p> <div class="mw-heading mw-heading3"><h3 id="Decimal_fractions_and_percentages">Decimal fractions and percentages</h3></div> <p>A <i><a href="/wiki/Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fraction</a></i> is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> to the right of a <a href="/wiki/Decimal_separator" title="Decimal separator">decimal separator</a>, the appearance of which (e.g., a period, an <a href="/wiki/Interpunct#In_mathematics_and_science" title="Interpunct">interpunct</a> (·), a comma) depends on the locale (for examples, see <i><a href="/wiki/Decimal_separator#Hindu–Arabic_numeral_system" title="Decimal separator">Decimal separator</a></i>). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the <a href="/wiki/Fractional_part" title="Fractional part">fractional part</a> of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠3<span class="sr-only">+</span><span class="tion"><span class="num">75</span><span class="sr-only">/</span><span class="den">100</span></span>⁠</span>. </p><p>Decimal fractions can also be expressed using <a href="/wiki/Scientific_notation" title="Scientific notation">scientific notation</a> with negative exponents, such as <span class="nowrap"><span data-sort-value="6993602299999999999♠"></span>6.023<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup></span>, which represents 0.0000006023. The <span class="nowrap"><span data-sort-value="6993100000000000000♠"></span>10<sup>−7</sup></span> represents a denominator of <span class="nowrap"><span data-sort-value="7007100000000000000♠"></span>10<sup>7</sup></span>. Dividing by <span class="nowrap"><span data-sort-value="7007100000000000000♠"></span>10<sup>7</sup></span> moves the decimal point 7 places to the left. </p><p>Decimal fractions with infinitely many digits to the right of the decimal separator represent an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>. 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">3</span></span>⁠</span> = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... </p><p>Another kind of fraction is the <a href="/wiki/Percentage" title="Percentage">percentage</a> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la">per centum</i>, meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100. </p><p>The related concept of <i><a href="/wiki/Permille" class="mw-redirect" title="Permille">permille</a></i> or <i>parts per thousand</i> (ppt) has an implied denominator of 1000, while the more general <a href="/wiki/Parts-per_notation" title="Parts-per notation">parts-per notation</a>, as in 75 <i>parts per million</i> (ppm), means that the proportion is 75/1,000,000. </p><p>Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By <a href="/wiki/Mental_calculation" title="Mental calculation">mental calculation</a>, it is easier to <a href="/wiki/Multiply" class="mw-redirect" title="Multiply">multiply</a> 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more <a href="/wiki/Accurate" class="mw-redirect" title="Accurate">accurate</a> to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6" (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6. </p> <div class="mw-heading mw-heading3"><h3 id="Mixed_numbers">Mixed numbers</h3></div> <p>A <b>mixed number</b> (also called a <i>mixed fraction</i> or <i>mixed numeral</i>) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or <i>concatenation</i>) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them. </p><p>As a basic example, two entire cakes and three quarters of another cake might be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de1a98a9929c24ee6345222f1e8c83c1c425f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.821ex; height:3.509ex;" alt="{\displaystyle 2{\tfrac {3}{4}}}"></span> cakes 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 2\ \,3/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mtext> </mtext> <mspace width="thinmathspace" /> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ \,3/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca1b9c61cf9025cd8bfe38acb723f547180427b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle 2\ \,3/4}"></span> cakes, with the numeral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> representing the whole cakes and the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"></span> representing the additional partial cake juxtaposed; this is more concise than the more explicit notation <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+{\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+{\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2171ffcdb7d37aff4303e27ab6392f2e2e40c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.661ex; height:3.509ex;" alt="{\displaystyle 2+{\tfrac {3}{4}}}"></span> cakes. The mixed number <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠2<span class="sr-only">+</span><span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> is pronounced "two and three quarters", with the integer and fraction portions connected by the word <i>and</i>.<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> Subtraction or negation is applied to the entire mixed numeral, so <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{\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2{\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc11fb59ed22f2e8c6396c79b87e2bee45fa9024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.629ex; height:3.509ex;" alt="{\displaystyle -2{\tfrac {3}{4}}}"></span> means <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 -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6997c9f9f06e100634b807929409759d514a3a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.246ex; height:3.509ex;" alt="{\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.}"></span> </p><p>Any mixed number can be converted to an <a href="/wiki/Improper_fraction" class="mw-redirect" title="Improper fraction">improper fraction</a> by applying the rules of <a href="#Adding_unlike_quantities">adding unlike quantities</a>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7303445a5982acb316ffa51b2420d1c142ea18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.141ex; height:3.509ex;" alt="{\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.}"></span> Conversely, an improper fraction can be converted to a mixed number using <a href="/wiki/Division_with_remainder" class="mw-redirect" title="Division with remainder">division with remainder</a>, with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left 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 {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802cf27ff5d8b2afe4b18b3dd7494a6e1a0755eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.886ex; height:3.509ex;" alt="{\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.}"></span> </p><p>In primary school, teachers often insist that every fractional result should be expressed as a mixed number.<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> Outside school, mixed numbers are commonly used for describing measurements, for instance <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠2<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> hours or 5 3/16 <a href="/wiki/Inch" title="Inch">inches</a>, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized <a href="/wiki/Metric_system" title="Metric system">metric system</a>. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a <a href="/wiki/Rational_number" title="Rational number">rational number</a>, the quotient <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>p</i></span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span> of integers, leaving behind the concepts of "improper fraction" and "mixed number".<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> College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expressions</a> means multiplication.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Historical_notions">Historical notions</h3></div> <div class="mw-heading mw-heading4"><h4 id="Egyptian_fraction">Egyptian fraction</h4></div> <p>An <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</a> is the sum of distinct positive unit fractions, for example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c57825b3992ab0d1bddf17c6c68baa9dc13104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.157ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}}"></span>. This definition derives from the fact that the <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">ancient Egyptians</a> expressed all fractions except <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"></span> in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73779cafebb46314575d513d3c8d23ca2e2f150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{7}}}"></span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>21</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027592e80dcfcc7338bea77451f9ac8132d2500e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.124ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.}"></span> Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {13}{17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>17</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {13}{17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8accb3400cc5f44e4d2705d2dde5c0a66faa0f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {13}{17}}}"></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>68</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027a26853a401f7e535a01ad1552cf9bdecab412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.477ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>68</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d35d8172860d2d530024d778871d98c3f1558f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.976ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Complex_and_compound_fractions">Complex and compound fractions</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">Complex numbers</a>.</div> <p>In a <b>complex fraction</b>, either the numerator, or the denominator, or both, is a fraction or a mixed number,<sup id="cite_ref-Trotter_23-0" class="reference"><a href="#cite_note-Trotter-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Barlow_24-0" class="reference"><a href="#cite_note-Barlow-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> corresponding to division of fractions. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1/2}{1/3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1/2}{1/3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739eccf8f07d2504545a4fb644277ac39f7de0e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.302ex; height:4.843ex;" alt="{\displaystyle {\tfrac {1/2}{1/3}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mn>26</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4400440f1d3ce06d187558123d68abbb78b3c2e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.781ex; height:3.509ex;" alt="{\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26}"></span> are complex fractions. To interpret nested fractions written "stacked" with a horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at <a href="#Division">§ Division</a>. For example: </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 {\begin{aligned}{\frac {\;\!{\tfrac {1}{2}}\;\!}{\tfrac {1}{3}}}&={\frac {1}{2}}\div {\frac {1}{3}}={\frac {1}{2}}\times {\frac {3}{1}}={\frac {3}{2}},\qquad {\frac {\;\!{\tfrac {3}{2}}\;\!}{5}}={\frac {3}{2}}\div 5={\frac {3}{2}}\times {\frac {1}{5}}={\frac {3}{10}},\\[10mu]{\frac {12{\tfrac {3}{4}}}{26}}&={\frac {12\times 4+3}{4}}\div 26={\frac {12\times 4+3}{4}}\times {\frac {1}{26}}={\frac {51}{104}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.856em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>÷</mo> <mn>5</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mrow> <mn>26</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>÷</mo> <mn>26</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>26</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>51</mn> <mn>104</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\;\!{\tfrac {1}{2}}\;\!}{\tfrac {1}{3}}}&={\frac {1}{2}}\div {\frac {1}{3}}={\frac {1}{2}}\times {\frac {3}{1}}={\frac {3}{2}},\qquad {\frac {\;\!{\tfrac {3}{2}}\;\!}{5}}={\frac {3}{2}}\div 5={\frac {3}{2}}\times {\frac {1}{5}}={\frac {3}{10}},\\[10mu]{\frac {12{\tfrac {3}{4}}}{26}}&={\frac {12\times 4+3}{4}}\div 26={\frac {12\times 4+3}{4}}\times {\frac {1}{26}}={\frac {51}{104}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ad62830598f9984928991e1bc41ec77a2b8235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:65.172ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\;\!{\tfrac {1}{2}}\;\!}{\tfrac {1}{3}}}&={\frac {1}{2}}\div {\frac {1}{3}}={\frac {1}{2}}\times {\frac {3}{1}}={\frac {3}{2}},\qquad {\frac {\;\!{\tfrac {3}{2}}\;\!}{5}}={\frac {3}{2}}\div 5={\frac {3}{2}}\times {\frac {1}{5}}={\frac {3}{10}},\\[10mu]{\frac {12{\tfrac {3}{4}}}{26}}&={\frac {12\times 4+3}{4}}\div 26={\frac {12\times 4+3}{4}}\times {\frac {1}{26}}={\frac {51}{104}}.\end{aligned}}}"></span></dd></dl> <p>A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5/10/20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5/10/20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff5cad15505cfca7253f6f9a6d824d4284fa46c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.137ex; height:2.843ex;" alt="{\displaystyle 5/10/20}"></span> could be plausibly interpreted as either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mn>20</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>40</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25308fa6d84313a6b93e6bea729711603acd8f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.727ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}}"></span> or as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{\big /}{\tfrac {10}{20}}=10.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>10.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{\big /}{\tfrac {10}{20}}=10.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5022d004de7a0c8c4835bc2bd0a42f1253af570a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.056ex; height:3.676ex;" alt="{\displaystyle 5{\big /}{\tfrac {10}{20}}=10.}"></span> The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (5/10){\big /}20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>10</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (5/10){\big /}20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd59980ee081966ee3826fb5c605eaee13b01559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.128ex; height:3.176ex;" alt="{\displaystyle (5/10){\big /}20}"></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 5{\big /}(10/20).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>20</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{\big /}(10/20).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afaebd81f55afd45ba11df0318b70ff83f1b6338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.774ex; height:3.176ex;" alt="{\displaystyle 5{\big /}(10/20).}"></span> </p><p>A <b>compound fraction</b> is a fraction of a fraction, or any number of fractions connected with the word <i>of</i>,<sup id="cite_ref-Trotter_23-1" class="reference"><a href="#cite_note-Trotter-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Barlow_24-1" class="reference"><a href="#cite_note-Barlow-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see <i><a href="#Multiplication">§ Multiplication</a></i>). For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73779cafebb46314575d513d3c8d23ca2e2f150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{7}}}"></span> is a compound fraction, corresponding to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>15</mn> <mn>28</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87dc90452902aa17eb3f89682ceccc0bb1345a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.735ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}"></span>. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd409db0f2718ae21b277b86759372fee37d450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.157ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}}"></span> is equivalent to the complex fraction <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3/4}{7/5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> <mrow> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3/4}{7/5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19980cc8c935fddf56467d78d7af8ef2a118a985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.302ex; height:4.843ex;" alt="{\displaystyle {\tfrac {3/4}{7/5}}}"></span>⁠</span>.) </p><p>Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated<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> and now used in no well-defined manner, partly even taken synonymously for each other<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> or for mixed numerals.<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> They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts". </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetic_with_fractions">Arithmetic with fractions</h2></div> <p>Like whole numbers, fractions obey the <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, and <a href="/wiki/Distributive_property" title="Distributive property">distributive</a> laws, and the rule against <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>. </p><p>Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_fractions">Equivalent fractions</h3></div> <p>Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3674532e6aafc9ab6f559abd113aa2d03ac7a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.009ex;" alt="{\displaystyle {\tfrac {n}{n}}}"></span> equals 1. Therefore, multiplying by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3674532e6aafc9ab6f559abd113aa2d03ac7a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.009ex;" alt="{\displaystyle {\tfrac {n}{n}}}"></span> is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span>⁠</span>. When the numerator and denominator are both multiplied by 2, the result is <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>, which has the same value (0.5) as <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>. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (<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>) make up half the cake (<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>). </p> <div class="mw-heading mw-heading4"><h4 id="Simplifying_(reducing)_fractions"><span id="Simplifying_.28reducing.29_fractions"></span>Simplifying (reducing) fractions <span class="anchor" id="Simplification"></span><span class="anchor" id="Reduction"></span></h4></div> <p>Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span> are divisible by <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>⁠</span>, then they can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=cd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=cd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719486aed0dd971dd2b5c1e38a6bd2b4d4f2bb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.551ex; height:2.176ex;" alt="{\displaystyle a=cd}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=ce}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=ce}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2344be710105630296b962e1687f87bcba48f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.186ex; height:2.176ex;" alt="{\displaystyle b=ce}"></span>, and the fraction becomes <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>cd</i></span><span class="sr-only">/</span><span class="den"><i>ce</i></span></span>⁠</span>, which can be reduced by dividing both the numerator and denominator by <i>c</i> to give the reduced fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i></span><span class="sr-only">/</span><span class="den"><i>e</i></span></span>⁠</span>. </p><p>If one takes for <span class="texhtml mvar" style="font-style:italic;">c</span> the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a>. One says that the fraction has been reduced to its <i><a href="/wiki/Lowest_terms" class="mw-redirect" title="Lowest terms">lowest terms</a></i>. </p><p>If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be <i><a href="/wiki/Irreducible_fraction" title="Irreducible fraction">irreducible</a></i>, <i>reduced</i>, or <i>in simplest terms</i>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>9</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b377f77b1e319507ab10bd4404b30a6192cb01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{9}}}"></span> is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4645a9398589aa1c62732b1a8f072b5ef17185d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{8}}}"></span> <i>is</i> in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. </p><p>Using these rules, we can show that <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> = <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> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">10</span><span class="sr-only">/</span><span class="den">20</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">50</span><span class="sr-only">/</span><span class="den">100</span></span>⁠</span></span>, for example. </p><p>As another example, since the greatest common divisor of 63 and 462 is 21, the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">63</span><span class="sr-only">/</span><span class="den">462</span></span>⁠</span> can be reduced to lowest terms by dividing the numerator and denominator by 21: </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 {63}{462}}={\frac {63\,\div \,21}{462\,\div \,21}}={\frac {3}{22}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>63</mn> <mn>462</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>63</mn> <mspace width="thinmathspace" /> <mo>÷</mo> <mspace width="thinmathspace" /> <mn>21</mn> </mrow> <mrow> <mn>462</mn> <mspace width="thinmathspace" /> <mo>÷</mo> <mspace width="thinmathspace" /> <mn>21</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>22</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {63}{462}}={\frac {63\,\div \,21}{462\,\div \,21}}={\frac {3}{22}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df420e57b6af5b811cf75c2884bd8fa5d32ef2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.591ex; height:5.343ex;" alt="{\displaystyle {\frac {63}{462}}={\frac {63\,\div \,21}{462\,\div \,21}}={\frac {3}{22}}.}"></span></dd></dl> <p>The <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> gives a method for finding the greatest common divisor of any two integers. </p> <div class="mw-heading mw-heading3"><h3 id="Comparing_fractions">Comparing fractions</h3></div> <p>Comparing fractions with the same positive denominator yields the same result as comparing the numerators: </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 {\tfrac {3}{4}}>{\tfrac {2}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}>{\tfrac {2}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380ed8a0f5ff34b303f19f4b6970ac9dfe6f466b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.415ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}>{\tfrac {2}{4}}}"></span> because <span class="nowrap">3 > 2</span>, and the equal denominators <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 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span> are positive.</dd></dl> <p>If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions: </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 {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> </mstyle> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> because </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mrow> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mi>b</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo>−</mo> <mn>3</mn> <mo><</mo> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bc3cb266f32d792a52d543e2c36047d6759784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.424ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.}"></span></dd></dl> <p>If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. </p><p>One way to compare fractions with different numerators and denominators is to find a common denominator. To compare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {c}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a2119495c75fea0bacd25e35ee40dc3897dffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.696ex; height:3.343ex;" alt="{\displaystyle {\tfrac {c}{d}}}"></span>, these are converted to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a\cdot d}{b\cdot d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>d</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>d</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a\cdot d}{b\cdot d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3424909e38999b279023941a2d0cc93b15606f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.023ex; height:3.843ex;" alt="{\displaystyle {\tfrac {a\cdot d}{b\cdot d}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {b\cdot c}{b\cdot d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>d</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {b\cdot c}{b\cdot d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76ef54c215cfbd4962d3b422512e75c2e408463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.859ex; height:3.843ex;" alt="{\displaystyle {\tfrac {b\cdot c}{b\cdot d}}}"></span> (where the dot signifies multiplication and is an alternative symbol to ×). Then <i>bd</i> is a common denominator and the numerators <i>ad</i> and <i>bc</i> can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare <i>ad</i> and <i>bc</i>, without evaluating <i>bd</i>, e.g., comparing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span> ? <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/116bcb43f4e1644702f1da86197efd0b6fefe07a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.415ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}}"></span>. </p><p>For the more laborious question <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{18}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>18</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{18}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97610f33e1cd083d26bb047f80ce45fcba46020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{18}}}"></span> ? <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{17}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>17</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{17}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ae99034ff5926d852050b4a01a2cc811f9ea7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.127ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{17}},}"></span> multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5\times 17}{18\times 17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mo>×</mo> <mn>17</mn> </mrow> <mrow> <mn>18</mn> <mo>×</mo> <mn>17</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5\times 17}{18\times 17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/321a67ef2c0d390212e0bf21c163a67304285569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.403ex; height:3.843ex;" alt="{\displaystyle {\tfrac {5\times 17}{18\times 17}}}"></span> ? <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {18\times 4}{18\times 17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>18</mn> <mo>×</mo> <mn>4</mn> </mrow> <mrow> <mn>18</mn> <mo>×</mo> <mn>17</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {18\times 4}{18\times 17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b3f7eaf84f54925f3903a16a8fd44011d2a709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.403ex; height:3.843ex;" alt="{\displaystyle {\tfrac {18\times 4}{18\times 17}}}"></span>. It is not necessary to calculate <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 18\times 17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>18</mn> <mo>×</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18\times 17}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2883151327b2f9b3a2775aa531674362705e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.49ex; height:2.176ex;" alt="{\displaystyle 18\times 17}"></span> – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>18</mn> </mfrac> </mstyle> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>17</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808b23c475ad0ff9bcc5441f59493be5030720dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.059ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}"></span>⁠</span>. </p><p>Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions. </p> <div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3></div> <p>The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows: </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 {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f1726a2cf4b7fca4f0830aa613bb3024639dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.832ex; height:3.509ex;" alt="{\displaystyle {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}}"></span>.</dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cake_fractions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cake_fractions.svg/270px-Cake_fractions.svg.png" decoding="async" width="270" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cake_fractions.svg/405px-Cake_fractions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cake_fractions.svg/540px-Cake_fractions.svg.png 2x" data-file-width="504" data-file-height="383" /></a><figcaption>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 {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span> of a cake is to be added to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4825cd2a1ca51dfc4d53042434f6d3733370a57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}}"></span> of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Adding_unlike_quantities">Adding unlike quantities</h4></div> <p>To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the <a href="#Reciprocals_and_the_"invisible_denominator"">invisible denominator</a> 1. </p><p>For adding quarters to thirds, both types of fraction are converted to twelfths, thus: </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}{4}}+{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}+{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}+{\frac {4}{12}}={\frac {7}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>×</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>×</mo> <mn>4</mn> </mrow> <mrow> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>12</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}+{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}+{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}+{\frac {4}{12}}={\frac {7}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45a8c186dc2703088da64a9e37a8e58f69326e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.947ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{4}}+{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}+{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}+{\frac {4}{12}}={\frac {7}{12}}.}"></span></dd></dl> <p>Consider adding the following two quantities: </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 {3}{5}}+{\frac {2}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/337c4ca40832f072063162ce54d8612042a943e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.484ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}.}"></span></dd></dl> <p>First, convert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f5356bc6141848e00fa630e4e1443f5d6fd2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{5}}}"></span> into fifteenths by multiplying both the numerator and denominator by three: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8771a7ef2a9ef0d9b4b6fadaf61aaf1b3757dfc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.735ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}}"></span>⁠</span>. Since <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> equals 1, multiplication by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> does not change the value of the fraction. </p><p>Second, convert <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">3</span></span>⁠</span> into fifteenths by multiplying both the numerator and denominator by five: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b2a9d8f45f40ed03bc9f64c3c1cbcf6755e455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.735ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}}"></span>⁠</span>. </p><p>Now it can be seen that </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 {3}{5}}+{\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c0d0adc8ca2b66e38daacd6b95d0464379ab19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.838ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}}"></span></dd></dl> <p>is equivalent to </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 {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>15</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>15</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192b17e71ce1a48076199d9ae45054aab4838b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.491ex; height:5.176ex;" alt="{\displaystyle {\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}.}"></span></dd></dl> <p>This method can be expressed algebraically: </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 {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}.}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>c</mi> <mi>b</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff275e908a7eda0b5fe307a2d622cf101a5be23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.83ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}.}"></span></dd></dl> <p>This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5558496705b3e588a303eeb875e6bc1bddda920a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {5}{6}}}"></span> the single denominators have a common factor 2, and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator. </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 {\begin{alignedat}{3}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot 6}{4\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {20}{24}}&&={\frac {19}{12}}\\[10mu]&={\frac {3\cdot 3}{4\cdot 3}}+{\frac {2\cdot 5}{2\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&&={\frac {19}{12}}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="0.856em 0.3em" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>6</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>24</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>20</mn> <mn>24</mn> </mfrac> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>12</mn> </mfrac> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot 6}{4\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {20}{24}}&&={\frac {19}{12}}\\[10mu]&={\frac {3\cdot 3}{4\cdot 3}}+{\frac {2\cdot 5}{2\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&&={\frac {19}{12}}.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0636adcc42216c77d9a0370ab4ebade53b8c0c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:42.375ex; height:12.176ex;" alt="{\displaystyle {\begin{alignedat}{3}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot 6}{4\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {20}{24}}&&={\frac {19}{12}}\\[10mu]&={\frac {3\cdot 3}{4\cdot 3}}+{\frac {2\cdot 5}{2\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&&={\frac {19}{12}}.\end{alignedat}}}"></span></dd></dl> <p>The smallest possible denominator is given by the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. This is called the least common denominator. </p> <div class="mw-heading mw-heading3"><h3 id="Subtraction">Subtraction</h3></div> <p>The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, </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 {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a1c27eb1cbd7fbe9e7e0cbc6ada7d12022a9fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.815ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}.}"></span></dd></dl> <p>To subtract a mixed number, an extra one can be borrowed from the minuend, for instance </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 4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98a999aa4631cd40bda4a45a07b6a10cbd79bf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.096ex; height:3.509ex;" alt="{\displaystyle 4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3></div> <div class="mw-heading mw-heading4"><h4 id="Multiplying_a_fraction_by_another_fraction">Multiplying a fraction by another fraction</h4></div> <p>To multiply fractions, multiply the numerators and multiply the denominators. Thus: </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 {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e17f5f593fceb5e9df9cd5d1313c1a9558e923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.744ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}.}"></span></dd></dl> <p>To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths. </p><p>A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example: </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 {2}{3}}\times {\frac {3}{4}}={\frac {{\color {RoyalBlue}{\cancel {\color {Black}2}}}^{~1}}{{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}}\times {\frac {{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}{{\color {RoyalBlue}{\cancel {\color {Black}4}}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0071BC"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mstyle mathcolor="#221E1F"> <mn>2</mn> </mstyle> </menclose> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F26035"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mstyle mathcolor="#221E1F"> <mn>3</mn> </mstyle> </menclose> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F26035"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mstyle mathcolor="#221E1F"> <mn>3</mn> </mstyle> </menclose> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0071BC"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mstyle mathcolor="#221E1F"> <mn>4</mn> </mstyle> </menclose> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {{\color {RoyalBlue}{\cancel {\color {Black}2}}}^{~1}}{{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}}\times {\frac {{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}{{\color {RoyalBlue}{\cancel {\color {Black}4}}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1ac234c23163744968111603f55ee42f87f245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:38.278ex; height:8.509ex;" alt="{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {{\color {RoyalBlue}{\cancel {\color {Black}2}}}^{~1}}{{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}}\times {\frac {{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}{{\color {RoyalBlue}{\cancel {\color {Black}4}}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}.}"></span></dd></dl> <p>A two is a common <a href="/wiki/Divisor" title="Divisor">factor</a> in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both. </p> <div class="mw-heading mw-heading4"><h4 id="Multiplying_a_fraction_by_a_whole_number">Multiplying a fraction by a whole number</h4></div> <p>Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. For example, </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 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>6</mn> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>18</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa49fddee4338bc2c97bed7f1ec32b937bd7ddcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.141ex; height:3.509ex;" alt="{\displaystyle 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}.}"></span></dd></dl> <p>This method works because the fraction 6/1 means six equal parts, each one of which is a whole. </p> <div class="mw-heading mw-heading4"><h4 id="Multiplying_mixed_numbers">Multiplying mixed numbers</h4></div> <p>The product of mixed numbers can be computed by converting each to an improper fraction.<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> For example: </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 3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>1</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>33</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8cb5502307bfe7de4971db009e87cf98d14394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.271ex; height:5.176ex;" alt="{\displaystyle 3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}.}"></span></dd></dl> <p>Alternately, mixed numbers can be treated as sums, and <a href="/wiki/FOIL_method" title="FOIL method">multiplied as binomials</a>. In this example, </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 3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9451a73b567c3e2c256bbb042bcdd36abc0ed244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.276ex; height:5.176ex;" alt="{\displaystyle 3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Division">Division</h3></div> <p>To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {10}{3}}\div 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {10}{3}}\div 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ec842509e23ad2e5f13e489ae4f60401b90100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.483ex; height:3.676ex;" alt="{\displaystyle {\tfrac {10}{3}}\div 5}"></span> equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span> and also equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>10</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205509f92c084bc36f38a12b315836bf83fd34ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.516ex; height:3.676ex;" alt="{\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}}"></span>, which reduces to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span>. To divide a number by a fraction, multiply that number by the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of that fraction. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\times {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>4</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\times {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7136b454549841138d898a338c3ababb8fda67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.204ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\times {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Converting_between_decimals_and_fractions">Converting between decimals and fractions</h3></div> <p>To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change <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">4</span></span>⁠</span> to a decimal, divide <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1.00</span> by <span class="nowrap"><span data-sort-value="7000400000000000000♠"></span>4</span> ("<span class="nowrap"><span data-sort-value="7000400000000000000♠"></span>4</span> into <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1.00</span>"), to obtain <span class="nowrap"><span data-sort-value="6999250000000000000♠"></span>0.25</span>. To change <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> to a decimal, divide <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1.000...</span> by <span class="nowrap"><span data-sort-value="7000300000000000000♠"></span>3</span> ("<span class="nowrap"><span data-sort-value="7000300000000000000♠"></span>3</span> into <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1.000...</span>"), and stop when the desired accuracy is obtained, e.g., at <span class="nowrap"><span data-sort-value="7000400000000000000♠"></span>4</span> decimals with <span class="nowrap"><span data-sort-value="6999333300000000000♠"></span>0.3333</span>. The fraction <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">4</span></span>⁠</span> can be written exactly with two decimal digits, while the fraction <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> cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1</span> followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus <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 12.3456={\tfrac {123456}{10000}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12.3456</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>123456</mn> <mn>10000</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12.3456={\tfrac {123456}{10000}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be5d2bd604deeb493b36d04b5f6ebbae567a67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.135ex; height:3.843ex;" alt="{\displaystyle 12.3456={\tfrac {123456}{10000}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Converting_repeating_decimals_to_fractions">Converting repeating decimals to fractions</h4></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/Decimal_representation#Conversion_to_fraction" title="Decimal representation">Decimal representation § Conversion to fraction</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/Repeating_decimal" title="Repeating decimal">Repeating decimal</a></div> <p>Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. </p><p>A conventional way to indicate a repeating decimal is to place a bar (known as a <a href="/wiki/Vinculum_(symbol)" title="Vinculum (symbol)">vinculum</a>) over the digits that repeat, for example 0.<span style="text-decoration:overline;">789</span> = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example: </p> <dl><dd>0.<span style="text-decoration:overline;">5</span> = 5/9</dd> <dd>0.<span style="text-decoration:overline;">62</span> = 62/99</dd> <dd>0.<span style="text-decoration:overline;">264</span> = 264/999</dd> <dd>0.<span style="text-decoration:overline;">6291</span> = 6291/9999</dd></dl> <p>If <a href="/wiki/Leading_zero" title="Leading zero">leading zeros</a> precede the pattern, the nines are suffixed by the same number of <a href="/wiki/Trailing_zero" title="Trailing zero">trailing zeros</a>: </p> <dl><dd>0.0<span style="text-decoration:overline;">5</span> = 5/90</dd> <dd>0.000<span style="text-decoration:overline;">392</span> = 392/999000</dd> <dd>0.00<span style="text-decoration:overline;">12</span> = 12/9900</dd></dl> <p>If a non-repeating set of decimals precede the pattern (such as 0.1523<span style="text-decoration:overline;">987</span>), one may write the number as the sum of the non-repeating and repeating parts, respectively: </p> <dl><dd>0.1523 + 0.0000<span style="text-decoration:overline;">987</span></dd></dl> <p>Then, convert both parts to fractions, and add them using the methods described above: </p> <dl><dd>1523 / 10000 + 987 / 9990000 = 1522464 / 9990000</dd></dl> <p>Alternatively, algebra can be used, such as below: </p> <ol><li>Let <i>x</i> = the repeating decimal: <dl><dd><i>x</i> = 0.1523<span style="text-decoration:overline;">987</span></dd></dl></li> <li>Multiply both sides by the power of 10 just great enough (in this case 10<sup>4</sup>) to move the decimal point just before the repeating part of the decimal number: <dl><dd>10,000<i>x</i> = 1,523.<span style="text-decoration:overline;">987</span></dd></dl></li> <li>Multiply both sides by the power of 10 (in this case 10<sup>3</sup>) that is the same as the number of places that repeat: <dl><dd>10,000,000<i>x</i> = 1,523,987.<span style="text-decoration:overline;">987</span></dd></dl></li> <li>Subtract the two equations from each other (if <span class="nowrap"><i>a</i> = <i>b</i></span> and <span class="nowrap"><i>c</i> = <i>d</i></span>, then <span class="nowrap"><i>a</i> − <i>c</i> = <i>b</i> − <i>d</i></span>): <dl><dd>10,000,000<i>x</i> − 10,000<i>x</i> = 1,523,987.<span style="text-decoration:overline;">987</span> − 1,523.<span style="text-decoration:overline;">987</span></dd></dl></li> <li>Continue the subtraction operation to clear the repeating decimal: <dl><dd>9,990,000<i>x</i> = 1,523,987 − 1,523</dd> <dd><span style="visibility:hidden">9,990,000<i>x</i></span> = 1,522,464</dd></dl></li> <li>Divide both sides by 9,990,000 to represent <i>x</i> as a fraction <dl><dd><i>x</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1522464</span><span class="sr-only">/</span><span class="den">9990000</span></span>⁠</span></dd></dl></li></ol> <div class="mw-heading mw-heading2"><h2 id="Fractions_in_abstract_mathematics">Fractions in abstract mathematics</h2></div> <p>In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are <a href="/wiki/Well_defined" class="mw-redirect" title="Well defined">consistent and reliable</a>. Mathematicians define a fraction as an ordered pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> of <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69238caa691e75f448aae578c1d7d93e8c9840b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.905ex; height:2.676ex;" alt="{\displaystyle b\neq 0,}"></span> for which the operations <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> are defined as follows:<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> </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,b)+(c,d)=(ad+bc,bd)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb08e1dc972afb22c15aa59344b4b78c1ce8ba80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.81ex; height:2.843ex;" alt="{\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,}"></span></dd> <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,b)-(c,d)=(ad-bc,bd)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227c77fa504515ab6f2cea294b367631c040a29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.81ex; height:2.843ex;" alt="{\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,}"></span></dd> <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,b)\cdot (c,d)=(ac,bd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\cdot (c,d)=(ac,bd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ca812be2352d4e56dda37f3beab255db4a0288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.207ex; height:2.843ex;" alt="{\displaystyle (a,b)\cdot (c,d)=(ac,bd)}"></span></dd> <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,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>÷</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>with, additionally, </mtext> </mrow> <mi>c</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611a939cdcf18caad30b7a52813d05359ef78912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.83ex; height:2.843ex;" alt="{\displaystyle (a,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)}"></span></dd></dl> <p>These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined 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 {\begin{aligned}-(a,b)&=(-a,b)&&{\text{additive inverse fractions,}}\\&&&{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-1}&=(b,a)&&{\text{multiplicative inverse fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>additive inverse fractions,</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with </mtext> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> as additive unities, and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplicative inverse fractions, for </mtext> </mrow> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>with </mtext> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> as multiplicative unities</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-(a,b)&=(-a,b)&&{\text{additive inverse fractions,}}\\&&&{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-1}&=(b,a)&&{\text{multiplicative inverse fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c069efe8f65322d81f8cd22dfbaad3cfc485a7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:64.49ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}-(a,b)&=(-a,b)&&{\text{additive inverse fractions,}}\\&&&{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-1}&=(b,a)&&{\text{multiplicative inverse fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}}"></span></dd></dl> <p>Furthermore, the <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a>, specified 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 (a,b)\sim (c,d)\quad \iff \quad ad=bc,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <mi>a</mi> <mi>d</mi> <mo>=</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\sim (c,d)\quad \iff \quad ad=bc,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8188efa0107a46780c052aeecefdf222c501c075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.973ex; height:2.843ex;" alt="{\displaystyle (a,b)\sim (c,d)\quad \iff \quad ad=bc,}"></span></dd></dl> <p>is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> of fractions. Each fraction from one equivalence class may be considered as a <a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representative</a> for the whole class, and each whole class may be considered as one abstract fraction. This equivalence is preserved by the above defined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions </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,b)\sim (a',b')\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\sim (a',b')\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0944f711b1ebf99f9400dce775b5f312bf93992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.932ex; height:3.009ex;" alt="{\displaystyle (a,b)\sim (a',b')\quad }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad (c,d)\sim (c',d')\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad (c,d)\sim (c',d')\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd60fa1ae0ea3c94c9e150382e96dde4e2b63835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.247ex; height:3.009ex;" alt="{\displaystyle \quad (c,d)\sim (c',d')\quad }"></span> imply <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,b)+(c,d))\sim ((a',b')+(c',d'))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((a,b)+(c,d))\sim ((a',b')+(c',d'))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ceb30ffb29f410e12bb745ccb135edb2d69e959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.412ex; height:3.009ex;" alt="{\displaystyle ((a,b)+(c,d))\sim ((a',b')+(c',d'))}"></span></dd></dl></dd></dl> <p>and similarly for the other operations. </p><p>In the case of fractions of integers, the fractions <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">a</span><span class="sr-only">/</span><span class="den">b</span></span>⁠</span> with <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> and <span class="texhtml"><i>b</i> > 0</span> are often taken as uniquely determined representatives for their <i>equivalent</i> fractions, which are considered to be the <i>same</i> rational number. This way the fractions of integers make up the field of the rational numbers. </p><p>More generally, <i>a</i> and <i>b</i> may be elements of any <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> <i>R</i>, in which case a fraction is an element of the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of <i>R</i>. For example, <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in one indeterminate, with coefficients from some integral domain <i>D</i>, are themselves an integral domain, call it <i>P</i>. So for <i>a</i> and <i>b</i> elements of <i>P</i>, the generated <i>field of fractions</i> is the field of <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a> (also known as the field of <a href="/wiki/Rational_function" title="Rational function">rational functions</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_fractions">Algebraic fractions</h2></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/Algebraic_fraction" title="Algebraic fraction">Algebraic fraction</a></div> <p>An algebraic fraction is the indicated <a href="/wiki/Quotient" title="Quotient">quotient</a> of two <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expressions</a>. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3x}{x^{2}+2x-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3x}{x^{2}+2x-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9528916dc5bde960b37350dfadb11c5c549ebfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.555ex; height:5.676ex;" alt="{\displaystyle {\frac {3x}{x^{2}+2x-3}}}"></span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43257a00c4988330dd0dac5c43ee9ea3f7339661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.105ex; height:6.343ex;" alt="{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}"></span>⁠</span>. Algebraic fractions are subject to the same <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> properties as arithmetic fractions. </p><p>If the numerator and the denominator are <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, as in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3x}{x^{2}+2x-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3x}{x^{2}+2x-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9528916dc5bde960b37350dfadb11c5c549ebfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.555ex; height:5.676ex;" alt="{\displaystyle {\frac {3x}{x^{2}+2x-3}}}"></span>⁠</span>, the algebraic fraction is called a <i>rational fraction</i> (or <i>rational expression</i>). An <i>irrational fraction</i> is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43257a00c4988330dd0dac5c43ee9ea3f7339661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.105ex; height:6.343ex;" alt="{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}"></span>⁠</span>. </p><p>The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47050d05d2ea37637a3ee6a838cfcd9f9f10055c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.615ex; height:7.509ex;" alt="{\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}}"></span>⁠</span>, is called a <b>complex fraction</b>. </p><p>The field of rational numbers is the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of the integers, while the integers themselves are not a field but rather an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. Similarly, the <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a> with coefficients in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, <a href="/wiki/Radical_expression" class="mw-redirect" title="Radical expression">radical expressions</a> representing numbers, such as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sqrt {2}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {2}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/552dde2a4619073ad08218d7b58cf5a196349391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.176ex;" alt="{\displaystyle \textstyle {\sqrt {2}}/2}"></span>⁠</span>, are also rational fractions, as are a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a> such 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="{\textstyle \pi /2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi /2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf81dd86006a3a0a437a8b5dfba23420a803a2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.304ex; height:2.843ex;" alt="{\textstyle \pi /2,}"></span> since all 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 {2}},\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}},\pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbfc1f762a84d0d258a09848160eb7474e8fc83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.111ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}},\pi ,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> are <a href="/wiki/Real_number" title="Real number">real numbers</a>, and thus considered as coefficients. These same numbers, however, are not rational fractions with <i>integer</i> coefficients. </p><p>The term <a href="/wiki/Partial_fraction" class="mw-redirect" title="Partial fraction">partial fraction</a> is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction <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 {2x}{x^{2}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x}{x^{2}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e6ed84595b2b30bd9fb73292da7a5881b1ed1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.223ex; height:5.676ex;" alt="{\displaystyle {\frac {2x}{x^{2}-1}}}"></span> can be decomposed as the sum of two fractions: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48ee812ea01ea80cd57631b6c4377edb1156f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.178ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}}"></span>⁠</span>. This is useful for the computation of <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> of <a href="/wiki/Rational_function" title="Rational function">rational functions</a> (see <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a> for more). </p> <div class="mw-heading mw-heading2"><h2 id="Radical_expressions">Radical expressions</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Nth_root" title="Nth root">Nth root</a> and <a href="/wiki/Rationalization_(mathematics)" class="mw-redirect" title="Rationalization (mathematics)">Rationalization (mathematics)</a></div> <p>A fraction may also contain <a href="/wiki/Nth_root" title="Nth root">radicals</a> in the numerator or the denominator. If the denominator contains radicals, it can be helpful to <a href="/wiki/Rationalisation_(mathematics)" title="Rationalisation (mathematics)">rationalize</a> it (compare <a href="/wiki/Nth_root#Simplified_form_of_a_radical_expression" title="Nth root">Simplified form of a radical expression</a>), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a <a href="/wiki/Monomial" title="Monomial">monomial</a> square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator: </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 {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msqrt> <mn>7</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <msqrt> <mn>7</mn> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>7</mn> </msqrt> <msqrt> <mn>7</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313ba8d603f1c4b7595f1b7c8537db77c0e3bb21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.423ex; height:6.843ex;" alt="{\displaystyle {\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}.}"></span></dd></dl> <p>The process of rationalization of <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a> denominators involves multiplying the top and the bottom of a fraction by the <a href="/wiki/Conjugate_(algebra)" class="mw-redirect" title="Conjugate (algebra)">conjugate</a> of the denominator so that the denominator becomes a rational number. For example: </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 {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mo>−</mo> <mn>20</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>11</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15443582ce45363a0f1dcfc80fd574e8d759982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:78.016ex; height:7.176ex;" alt="{\displaystyle {\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}},}"></span></dd> <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 {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mo>−</mo> <mn>20</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>11</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449d424db3f717f77e03f628b422339729d32c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:78.016ex; height:7.176ex;" alt="{\displaystyle {\frac {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}.}"></span></dd></dl> <p>Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator. </p> <div class="mw-heading mw-heading2"><h2 id="Typographical_variations">Typographical variations</h2></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/Slash_(punctuation)#Encoding" title="Slash (punctuation)">Slash § Encoding</a></div> <p>In computer displays and <a href="/wiki/Typography" title="Typography">typography</a>, simple fractions are sometimes printed as a single character, e.g. ½ (<a href="/wiki/One_half" title="One half">one half</a>). See the article on <a href="/wiki/Number_Forms" title="Number Forms">Number Forms</a> for information on doing this in <a href="/wiki/Unicode" title="Unicode">Unicode</a>. </p><p>Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:<sup id="cite_ref-galen_30-0" class="reference"><a href="#cite_note-galen-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <ul><li><b>Special fractions</b>: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts.</li> <li><b>Case fractions</b>: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them <i>upright</i>. An example would be <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>, but rendered with the same height as other characters. Some sources include all rendering of fractions as <i>case fractions</i> if they take only one typographical space, regardless of the direction of the bar.<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></li> <li><b>Shilling</b> or <b>solidus fractions</b>: 1/2, so called because this notation was used for pre-decimal British currency (<a href="/wiki/%C2%A3sd" title="£sd">£sd</a>), as in "2/6" for a <a href="/wiki/Half_crown_(British_coin)" title="Half crown (British coin)">half crown</a>, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions (<a href="#Complex_fractions">complex fractions</a>) or within exponents to increase legibility. Fractions written this way, also known as <b>piece fractions</b>,<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> are written all on one typographical line, but take 3 or more typographical spaces.</li> <li><b>Built-up fractions</b>: <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}{2}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"></span>. This notation uses two or more lines of ordinary text and results in a variation in spacing between lines when included within other text. While large and legible, these can be disruptive, particularly for simple fractions or within complex fractions.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></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 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 History 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/Fraction" title="Special:EditPage/Fraction">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 History section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Fraction%22">"Fraction"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Fraction%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Fraction%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Fraction%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Fraction%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Fraction%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">June 2023</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 earliest fractions were <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of <a href="/wiki/Integer" title="Integer">integers</a>: ancient symbols representing one part of two, one part of three, one part of four, and so on.<sup id="cite_ref-eves_33-0" class="reference"><a href="#cite_note-eves-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The Egyptians used <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fractions</a> <span title="circa">c.</span><span style="white-space:nowrap;"> 1000</span> BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with <a href="/wiki/Unit_fraction" title="Unit fraction">unit fractions</a>. Their methods gave the same answer as modern methods.<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> The Egyptians also had a different notation for <a href="/wiki/Dyadic_fraction" class="mw-redirect" title="Dyadic fraction">dyadic fractions</a>, used for certain systems of weights and measures.<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> </p><p>The <a href="/wiki/Ancient_Greece" title="Ancient Greece">Greeks</a> used unit fractions and (later) <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fractions</a>. <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Followers</a> of the <a href="/wiki/Ancient_Greece" title="Ancient Greece">Greek</a> <a href="/wiki/Greek_philosophy" class="mw-redirect" title="Greek philosophy">philosopher</a> <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> 530</span> BC) discovered that the <a href="/wiki/Square_root_of_two" class="mw-redirect" title="Square root of two">square root of two</a> <a href="/wiki/Irrational_numbers" class="mw-redirect" title="Irrational numbers">cannot be expressed as a fraction of integers</a>. (This is commonly though probably erroneously ascribed to <a href="/wiki/Hippasus" title="Hippasus">Hippasus</a> of <a href="/wiki/Metapontum" title="Metapontum">Metapontum</a>, who is said to have been executed for revealing this fact.) In <span class="nowrap">150 BC</span> <a href="/wiki/Jain" class="mw-redirect" title="Jain">Jain</a> mathematicians in <a href="/wiki/History_of_India" title="History of India">India</a> wrote the "<a href="/wiki/Sthananga_Sutra" title="Sthananga Sutra">Sthananga Sutra</a>", which contains work on the theory of numbers, arithmetical operations, and operations with fractions. </p><p>A modern expression of fractions known as <b>bhinnarasi</b> seems to have originated in India in the work of <a href="/wiki/Aryabhatta" class="mw-redirect" title="Aryabhatta">Aryabhatta</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> AD 500</span>),<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. (February 2016)">citation needed</span></a></i>]</sup> <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> 628</span>), and <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhaskara</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> 1150</span>).<sup id="cite_ref-jeff_36-0" class="reference"><a href="#cite_note-jeff-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Their works form fractions by placing the numerators (<a href="/wiki/Sanskrit_language" class="mw-redirect" title="Sanskrit language">Sanskrit</a>: <i lang="sa">amsa</i>) over the denominators (<span title="Sanskrit-language text"><i lang="sa">cheda</i></span>), but without a bar between them.<sup id="cite_ref-jeff_36-1" class="reference"><a href="#cite_note-jeff-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Sanskrit_literature" title="Sanskrit literature">Sanskrit literature</a>, fractions were always expressed as an addition to or subtraction from an integer.<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. (February 2016)">citation needed</span></a></i>]</sup> The integer was written on one line and the fraction in its two parts on the next line. If the fraction was marked by a small circle <span class="nowrap">⟨०⟩</span> or cross <span class="nowrap">⟨+⟩</span>, it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, <a href="/wiki/Bhaskara_I" class="mw-redirect" title="Bhaskara I">Bhaskara I</a> writes:<sup id="cite_ref-filliozat-p152_37-0" class="reference"><a href="#cite_note-filliozat-p152-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>६  १  २</dd> <dd>१  १  १<sub>०</sub></dd> <dd>४  ५  ९</dd></dl> <p>which is the equivalent of </p> <dl><dd>6  1  2</dd> <dd>1  1  −1</dd> <dd>4  5  9</dd></dl> <p>and would be written in modern notation as 6<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">4</span></span>⁠</span>, 1<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">5</span></span>⁠</span>, and 2 − <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">9</span></span>⁠</span> (i.e., 1<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span>). </p><p>The horizontal fraction bar is first attested in the work of <a href="/wiki/Al-Hass%C4%81r" class="mw-redirect" title="Al-Hassār">Al-Hassār</a> (<abbr title="floruit ('flourished' – known to have been active at a particular time or during a particular period)">fl.</abbr><span style="white-space:nowrap;"> 1200</span>),<sup id="cite_ref-jeff_36-2" class="reference"><a href="#cite_note-jeff-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Muslim mathematician</a> from <a href="/wiki/Fes" class="mw-redirect" title="Fes">Fez</a>, <a href="/wiki/Morocco" title="Morocco">Morocco</a>, who specialized in <a href="/wiki/Islamic_inheritance_jurisprudence" title="Islamic inheritance jurisprudence">Islamic inheritance jurisprudence</a>. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3\quad 1}{5\quad 3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mspace width="1em" /> <mn>1</mn> </mrow> <mrow> <mn>5</mn> <mspace width="1em" /> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\quad 1}{5\quad 3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078d004ad0c501c44d3ff1d2360ade80bb2fad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.484ex; height:5.176ex;" alt="{\displaystyle {\frac {3\quad 1}{5\quad 3}}}"></span>".</span><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> The same fractional notation—with the fraction given before the integer<sup id="cite_ref-jeff_36-3" class="reference"><a href="#cite_note-jeff-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>—appears soon after in the work of <a href="/wiki/Leonardo_Fibonacci" class="mw-redirect" title="Leonardo Fibonacci">Leonardo Fibonacci</a> in the 13th century.<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>In discussing the origins of <a href="/wiki/Decimal_fractions" class="mw-redirect" title="Decimal fractions">decimal fractions</a>, <a href="/wiki/Dirk_Jan_Struik" title="Dirk Jan Struik">Dirk Jan Struik</a> states:<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> <blockquote> <p>The introduction of decimal fractions as a common computational practice can be dated back to the <a href="/wiki/Flemish_Region" title="Flemish Region">Flemish</a> pamphlet <i>De Thiende</i>, published at <a href="/wiki/Leiden" title="Leiden">Leyden</a> in 1585, together with a French translation, <i>La Disme</i>, by the Flemish mathematician <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> (1548–1620), then settled in the Northern <a href="/wiki/Netherlands" title="Netherlands">Netherlands</a>. It is true that decimal fractions were used by the <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese</a> many centuries before Stevin and that the Persian astronomer <a href="/wiki/Al-K%C4%81sh%C4%AB" class="mw-redirect" title="Al-Kāshī">Al-Kāshī</a> used both decimal and <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> fractions with great ease in his <i>Key to arithmetic</i> (<a href="/wiki/Samarkand" title="Samarkand">Samarkand</a>, early fifteenth century).<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <p>While the <a href="/wiki/Persian_people" class="mw-redirect" title="Persian people">Persian</a> mathematician <a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the <a href="/wiki/Baghdad" title="Baghdad">Baghdadi</a> mathematician <a href="/wiki/Abu%27l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a> as early as the 10th century.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>n 2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_formal_education">In formal education</h2></div> <div class="mw-heading mw-heading3"><h3 id="Primary_schools">Primary schools</h3></div> <p>In <a href="/wiki/Primary_school" title="Primary school">primary schools</a>, fractions have been demonstrated through <a href="/wiki/Cuisenaire_rods" title="Cuisenaire rods">Cuisenaire rods</a>, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), <a href="/wiki/Pattern_block" class="mw-redirect" title="Pattern block">pattern blocks</a>, pie-shaped pieces, plastic rectangles, grid paper, <a href="/w/index.php?title=Dot_paper&action=edit&redlink=1" class="new" title="Dot paper (page does not exist)">dot paper</a>, <a href="/wiki/Geoboard" title="Geoboard">geoboards</a>, counters and computer software. </p> <div class="mw-heading mw-heading3"><h3 id="Documents_for_teachers">Documents for teachers</h3></div> <p>Several states in the United States have adopted learning trajectories from the <a href="/wiki/Common_Core_State_Standards_Initiative" class="mw-redirect" title="Common Core State Standards Initiative">Common Core State Standards Initiative</a>'s guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></span><span class="sr-only">/</span><span class="den"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></span></span>⁠</span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is a whole number and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is a positive whole number. (The word <i>fraction</i> in these standards always refers to a non-negative number.)"<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The document itself also refers to negative fractions. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Cross_multiplication" class="mw-redirect" title="Cross multiplication">Cross multiplication</a></li> <li><a href="/wiki/0.999..." title="0.999...">0.999...</a></li> <li><a href="/wiki/Multiple_(mathematics)" title="Multiple (mathematics)">Multiple</a></li> <li><a href="/wiki/FRACTRAN" title="FRACTRAN">FRACTRAN</a></li></ul> <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 class="mw-selflink selflink">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-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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Some typographers such as <a href="/wiki/Robert_Bringhurst" title="Robert Bringhurst">Bringhurst</a> mistakenly distinguish the slash <span class="nowrap">⟨<a href="/wiki//" class="mw-redirect" title="/">/</a>⟩</span> as the <i><a href="https://en.wiktionary.org/wiki/virgule" class="extiw" title="wikt:virgule">virgule</a></i> and the fraction slash <span class="nowrap">⟨<a href="/wiki/%E2%81%84" class="mw-redirect" title="⁄">⁄</a>⟩</span> as the <i><a href="/wiki/Solidus_mark" class="mw-redirect" title="Solidus mark">solidus</a></i>,<sup id="cite_ref-bringhurst_6-0" class="reference"><a href="#cite_note-bringhurst-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> although in fact both are synonyms for the standard slash.<sup id="cite_ref-verg_7-0" class="reference"><a href="#cite_note-verg-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-oedsolid_8-0" class="reference"><a href="#cite_note-oedsolid-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">While there is some disagreement among history of mathematics scholars as to the primacy of al-Uqlidisi's contribution, there is no question as to his major contribution to the concept of decimal fractions.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <p><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein2003" class="citation encyclopaedia cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric</a> (2003). "CRC Concise Encyclopedia of Mathematics, Second Edition". <i>CRC Concise Encyclopedia of Mathematics</i>. Chapman & Hall/CRC. p. 1925. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-347-2" title="Special:BookSources/1-58488-347-2"><bdi>1-58488-347-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=CRC+Concise+Encyclopedia+of+Mathematics%2C+Second+Edition&rft.btitle=CRC+Concise+Encyclopedia+of+Mathematics&rft.pages=1925&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2003&rft.isbn=1-58488-347-2&rft.aulast=Weisstein&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span> </p> <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">H. Wu, "The Mis-Education of Mathematics Teachers", <i>Notices of the American Mathematical Society</i>, Volume 58, Issue 03 (March 2011), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201103/rtx110300372p.pdf#page374">p. 374</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170820101254/http://www.ams.org/notices/201103/rtx110300372p.pdf#page374">Archived</a> 2017-08-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-schwartzman-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-schwartzman_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartzman1994" class="citation book cs1">Schwartzman, Steven (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/wordsofmathemati0000schw"><i>The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English</i></a></span>. Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-511-9" title="Special:BookSources/978-0-88385-511-9"><bdi>978-0-88385-511-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Words+of+Mathematics%3A+An+Etymological+Dictionary+of+Mathematical+Terms+Used+in+English&rft.pub=Mathematical+Association+of+America&rft.date=1994&rft.isbn=978-0-88385-511-9&rft.aulast=Schwartzman&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fwordsofmathemati0000schw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/fractions.html">"Fractions"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Fractions&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Ffractions.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-ambrose-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-ambrose_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ambrose_4-1"><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="CITEREFAmbrose_&_al." class="citation book cs1">Ambrose, Gavin; et al. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IW9MAQAAQBAJ"><i>The Fundamentals of Typography</i></a> (2nd ed.). Lausanne: AVA Publishing. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IW9MAQAAQBAJ&pg=PA74">74</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-940411-76-4" title="Special:BookSources/978-2-940411-76-4"><bdi>978-2-940411-76-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304022742/https://books.google.co.jp/books?id=IW9MAQAAQBAJ&printsec=frontcover">Archived</a> from the original on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fundamentals+of+Typography&rft.place=Lausanne&rft.pages=74&rft.edition=2nd&rft.pub=AVA+Publishing&rft.date=2006&rft.isbn=978-2-940411-76-4&rft.aulast=Ambrose&rft.aufirst=Gavin&rft.au=Paul+Harris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIW9MAQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori1928">Cajori (1928)</a>, "275. The solidus", pp. 312–314</span> </li> <li id="cite_note-bringhurst-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-bringhurst_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBringhurst2002" class="citation book cs1">Bringhurst, Robert (2002). "5.2.5: Use the Virgule with Words and Dates, the Solidus with Split-level Fractions". <i>The Elements of Typographic Style</i> (3rd ed.). <a href="/wiki/Point_Roberts,_Washington" title="Point Roberts, Washington">Point Roberts</a>: Hartley & Marks. pp. 81–82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88179-206-5" title="Special:BookSources/978-0-88179-206-5"><bdi>978-0-88179-206-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.2.5%3A+Use+the+Virgule+with+Words+and+Dates%2C+the+Solidus+with+Split-level+Fractions&rft.btitle=The+Elements+of+Typographic+Style&rft.place=Point+Roberts&rft.pages=81-82&rft.edition=3rd&rft.pub=Hartley+%26+Marks&rft.date=2002&rft.isbn=978-0-88179-206-5&rft.aulast=Bringhurst&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-verg-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-verg_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"virgule, <i>n.</i>". <i>Oxford English Dictionary</i> (1st ed.). Oxford: Oxford University Press. 1917.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=virgule%2C+n.&rft.btitle=Oxford+English+Dictionary&rft.place=Oxford&rft.edition=1st&rft.pub=Oxford+University+Press&rft.date=1917&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-oedsolid-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-oedsolid_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"solidus, <i>n.<sup>1</sup></i>". <i>Oxford English Dictionary</i> (1st ed.). Oxford: Oxford University Press. 1913.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=solidus%2C+n.%3Csup%3E1%3C%2Fsup%3E&rft.btitle=Oxford+English+Dictionary&rft.place=Oxford&rft.edition=1st&rft.pub=Oxford+University+Press&rft.date=1913&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEasterday1982" class="citation journal cs1">Easterday, Kenneth E. (Winter 1982). "One-hundred fifty years of vulgar fractions". <i>Contemporary Education</i>. <b>53</b> (2): 83–88. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ProQuest" title="ProQuest">ProQuest</a> <a rel="nofollow" class="external text" href="https://search.proquest.com/docview/1291644250">1291644250</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Contemporary+Education&rft.atitle=One-hundred+fifty+years+of+vulgar+fractions&rft.ssn=winter&rft.volume=53&rft.issue=2&rft.pages=83-88&rft.date=1982&rft.aulast=Easterday&rft.aufirst=Kenneth+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-Smith1958-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Smith1958_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Smith1958_11-1"><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="CITEREFDavid_E._Smith1958" class="citation book cs1">David E. Smith (1 June 1958). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC"><i>History of Mathematics</i></a>. Courier Corporation. p. 219. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-20430-7" title="Special:BookSources/978-0-486-20430-7"><bdi>978-0-486-20430-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pages=219&rft.pub=Courier+Corporation&rft.date=1958-06-01&rft.isbn=978-0-486-20430-7&rft.au=David+E.+Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuTytJGnTf1kC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></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="CITEREFPerryPerry1981" class="citation book cs1">Perry, Owen; Perry, Joyce (1981). "Chapter 2: Common fractions". <i>Mathematics I</i>. Palgrave Macmillan UK. pp. 13–25. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-349-05230-1_2">10.1007/978-1-349-05230-1_2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2%3A+Common+fractions&rft.btitle=Mathematics+I&rft.pages=13-25&rft.pub=Palgrave+Macmillan+UK&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2F978-1-349-05230-1_2&rft.aulast=Perry&rft.aufirst=Owen&rft.au=Perry%2C+Joyce&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-Williams2011-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Williams2011_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJack_Williams2011" class="citation book cs1">Jack Williams (19 November 2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dTqHIM1ds1kC&pg=PA87"><i>Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation</i></a>. Springer Science & Business Media. pp. 87–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85729-862-1" title="Special:BookSources/978-0-85729-862-1"><bdi>978-0-85729-862-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robert+Recorde%3A+Tudor+Polymath%2C+Expositor+and+Practitioner+of+Computation&rft.pages=87-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011-11-19&rft.isbn=978-0-85729-862-1&rft.au=Jack+Williams&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdTqHIM1ds1kC%26pg%3DPA87&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-Record1654-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Record1654_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRecord1654" class="citation book cs1">Record, Robert (1654). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=colv-l9SOlcC&pg=PA266"><i>Record's Arithmetick: Or, the Ground of Arts: Teaching the Perfect Work and Practise of Arithmetick ... Made by Mr. Robert Record ... Afterward Augmented by Mr. John Dee. And Since Enlarged with a Third Part of Rules of Practise ... By John Mellis. And Now Diligently Perused, Corrected ... and Enlarged; with an Appendix of Figurative Numbers ... with Tables of Board and Timber Measure ... the First Calculated by R. C. But Corrected, and the Latter ... Calculated by Ro. Hartwell ...</i></a> James Flesher, and are to be sold by Edward Dod. pp. 266–.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Record%27s+Arithmetick%3A+Or%2C+the+Ground+of+Arts%3A+Teaching+the+Perfect+Work+and+Practise+of+Arithmetick+...+Made+by+Mr.+Robert+Record+...+Afterward+Augmented+by+Mr.+John+Dee.+And+Since+Enlarged+with+a+Third+Part+of+Rules+of+Practise+...+By+John+Mellis.+And+Now+Diligently+Perused%2C+Corrected+...+and+Enlarged%3B+with+an+Appendix+of+Figurative+Numbers+...+with+Tables+of+Board+and+Timber+Measure+...+the+First+Calculated+by+R.+C.+But+Corrected%2C+and+the+Latter+...+Calculated+by+Ro.+Hartwell+...&rft.pages=266-&rft.pub=James+Flesher%2C+and+are+to+be+sold+by+Edward+Dod&rft.date=1654&rft.aulast=Record&rft.aufirst=Robert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dcolv-l9SOlcC%26pg%3DPA266&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaurel_BrennerPeterson2004" class="citation web cs1">Laurel Brenner; Peterson (31 March 2004). <a rel="nofollow" class="external text" href="http://mathforum.org/library/drmath/view/65128.html">"Ask Dr. Math: Can Negative Fractions Also Be Proper or Improper?"</a>. <i>Math Forum</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141109010850/http://mathforum.org/library/drmath/view/65128.html">Archived</a> from the original on 9 November 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-10-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Forum&rft.atitle=Ask+Dr.+Math%3A+Can+Negative+Fractions+Also+Be+Proper+or+Improper%3F&rft.date=2004-03-31&rft.au=Laurel+Brenner&rft.au=Peterson&rft_id=http%3A%2F%2Fmathforum.org%2Flibrary%2Fdrmath%2Fview%2F65128.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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/20120415053421/http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm">"Proper Fraction"</a>. <i>New England Compact Math Resources</i>. Archived from <a rel="nofollow" class="external text" href="http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm">the original</a> on 2012-04-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-12-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=New+England+Compact+Math+Resources&rft.atitle=Proper+Fraction&rft_id=http%3A%2F%2Fwww.necompact.org%2Fea%2Fgle_support%2FMath%2Fresources_number%2Fprop_fraction.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFGreer1986" class="citation book cs1">Greer, A. (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wX2dxeDahAwC&pg=PA5"><i>New comprehensive mathematics for 'O' level</i></a> (2nd ed., reprinted ed.). Cheltenham: Thornes. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85950-159-0" title="Special:BookSources/978-0-85950-159-0"><bdi>978-0-85950-159-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190119204758/https://books.google.com/books?id=wX2dxeDahAwC&pg=PA5">Archived</a> from the original on 2019-01-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-07-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+comprehensive+mathematics+for+%27O%27+level&rft.place=Cheltenham&rft.pages=5&rft.edition=2nd+ed.%2C+reprinted&rft.pub=Thornes&rft.date=1986&rft.isbn=978-0-85950-159-0&rft.aulast=Greer&rft.aufirst=A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwX2dxeDahAwC%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFKelley2004" class="citation book cs1">Kelley, W. Michael (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K1hCltk-2RwC&pg=PA25"><i>The Complete Idiot's Guide to Algebra</i></a>. Penguin. p. 25. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781592571611" title="Special:BookSources/9781592571611"><bdi>9781592571611</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+Complete+Idiot%27s+Guide+to+Algebra&rft.pages=25&rft.pub=Penguin&rft.date=2004&rft.isbn=9781592571611&rft.aulast=Kelley&rft.aufirst=W.+Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK1hCltk-2RwC%26pg%3DPA25&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFWingard-Nelson2014" class="citation book cs1">Wingard-Nelson, Rebecca (2014). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/readyforfraction0000wing/page/14/mode/2up?q=%22mixed+number+out+loud%22"><i>Ready for Fractions and Decimals</i></a></span>. Enslow. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7660-4247-6" title="Special:BookSources/978-0-7660-4247-6"><bdi>978-0-7660-4247-6</bdi></a>. <q>When you read a mixed number out loud, you say the whole number, the word <i>and</i>, then the fraction. The mixed number <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠2<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> is read as <i>two and one fourth</i>.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ready+for+Fractions+and+Decimals&rft.pages=14&rft.pub=Enslow&rft.date=2014&rft.isbn=978-0-7660-4247-6&rft.aulast=Wingard-Nelson&rft.aufirst=Rebecca&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Freadyforfraction0000wing%2Fpage%2F14%2Fmode%2F2up%3Fq%3D%2522mixed%2Bnumber%2Bout%2Bloud%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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 id="CITEREFWu2011" class="citation book cs1">Wu, Hung-Hsi (2011). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/understandingnum0000wuho/page/225/?q=%22mixed+number%22"><i>Understanding Numbers in Elementary School Mathematics</i></a></span>. American Mathematical Society. §14.3 Mixed Numbers, pp. 225–227. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5260-6" title="Special:BookSources/978-0-8218-5260-6"><bdi>978-0-8218-5260-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Numbers+in+Elementary+School+Mathematics&rft.pages=%C2%A714.3+Mixed+Numbers%2C+pp.-225-227&rft.pub=American+Mathematical+Society&rft.date=2011&rft.isbn=978-0-8218-5260-6&rft.aulast=Wu&rft.aufirst=Hung-Hsi&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funderstandingnum0000wuho%2Fpage%2F225%2F%3Fq%3D%2522mixed%2Bnumber%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFGardiner2016" class="citation book cs1">Gardiner, Tony (2016). <i>Teaching Mathematics at Secondary Level</i>. OBP Series in Mathematics. Open Book Publishers. p. 89. <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/10.11647%2FOBP.0071">10.11647/OBP.0071</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781783741373" title="Special:BookSources/9781783741373"><bdi>9781783741373</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teaching+Mathematics+at+Secondary+Level&rft.series=OBP+Series+in+Mathematics&rft.pages=89&rft.pub=Open+Book+Publishers&rft.date=2016&rft_id=info%3Adoi%2F10.11647%2FOBP.0071&rft.isbn=9781783741373&rft.aulast=Gardiner&rft.aufirst=Tony&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFLeeMessner2000" class="citation journal cs1">Lee, Mary A; Messner, Shelley J. (2000). "Analysis of concatenations and order of operations in written mathematics". <i>School Science and Mathematics</i>. <b>100</b> (4): 173–180. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1949-8594.2000.tb17254.x">10.1111/j.1949-8594.2000.tb17254.x</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ProQuest" title="ProQuest">ProQuest</a> <a rel="nofollow" class="external text" href="https://search.proquest.com/docview/195210281">195210281</a>. <q>College students have had many years of high school and perhaps college experience in which multiplication has been the implied operation in concatenations such as 4<i>x</i>, with little classroom experience with mixed numbers, so that for them, when returning to mixed number forms, they apply their recent knowledge of multiplication as the implied operation in concatenation to the 'new' situation of mixed numbers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=School+Science+and+Mathematics&rft.atitle=Analysis+of+concatenations+and+order+of+operations+in+written+mathematics&rft.volume=100&rft.issue=4&rft.pages=173-180&rft.date=2000&rft_id=info%3Adoi%2F10.1111%2Fj.1949-8594.2000.tb17254.x&rft.aulast=Lee&rft.aufirst=Mary+A&rft.au=Messner%2C+Shelley+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-Trotter-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-Trotter_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Trotter_23-1"><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="CITEREFTrotter1853" class="citation book cs1">Trotter, James (1853). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=a0sDAAAAQAAJ&q=%22complex+fraction%22&pg=PA65"><i>A complete system of arithmetic</i></a>. p. 65.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+complete+system+of+arithmetic&rft.pages=65&rft.date=1853&rft.aulast=Trotter&rft.aufirst=James&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Da0sDAAAAQAAJ%26q%3D%2522complex%2Bfraction%2522%26pg%3DPA65&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-Barlow-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Barlow_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Barlow_24-1"><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="CITEREFBarlow1814" class="citation book cs1">Barlow, Peter (1814). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BBowAAAAYAAJ&q=%2B%22complex+fraction%22+%2B%22compound+fraction%22&pg=PT329"><i>A new mathematical and philosophical dictionary</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+new+mathematical+and+philosophical+dictionary&rft.date=1814&rft.aulast=Barlow&rft.aufirst=Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBBowAAAAYAAJ%26q%3D%252B%2522complex%2Bfraction%2522%2B%252B%2522compound%2Bfraction%2522%26pg%3DPT329&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171201182513/https://www.collinsdictionary.com/dictionary/english/complex-fraction">"complex fraction"</a>. <i>Collins English Dictionary</i>. Archived from <a rel="nofollow" class="external text" href="https://www.collinsdictionary.com/dictionary/english/complex-fraction">the original</a> on 2017-12-01<span class="reference-accessdate">. Retrieved <span class="nowrap">29 August</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=complex+fraction&rft.btitle=Collins+English+Dictionary&rft_id=https%3A%2F%2Fwww.collinsdictionary.com%2Fdictionary%2Fenglish%2Fcomplex-fraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.collinsdictionary.com/dictionary/english/complex-fraction">"Complex fraction definition and meaning"</a>. <i>Collins English Dictionary</i>. 2018-03-09. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171201182513/https://www.collinsdictionary.com/dictionary/english/complex-fraction">Archived</a> from the original on 2017-12-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Complex+fraction+definition+and+meaning&rft.btitle=Collins+English+Dictionary&rft.date=2018-03-09&rft_id=https%3A%2F%2Fwww.collinsdictionary.com%2Fdictionary%2Fenglish%2Fcomplex-fraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.sosmath.com/algebra/fraction/frac5/frac5.html">"Compound Fractions"</a>. Sosmath.com. 1996-02-05. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180314105714/http://www.sosmath.com/algebra/fraction/frac5/frac5.html">Archived</a> from the original on 2018-03-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Compound+Fractions&rft.pub=Sosmath.com&rft.date=1996-02-05&rft_id=http%3A%2F%2Fwww.sosmath.com%2Falgebra%2Ffraction%2Ffrac5%2Ffrac5.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchoenbornSimkins2010" class="citation book cs1">Schoenborn, Barry; Simkins, Bradley (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/technical-math-for-dummies_202007/page/120">"8. Fun with Fractions"</a>. <i>Technical Math For Dummies</i>. Hoboken: <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">Wiley Publishing Inc.</a> p. 120. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-59874-0" title="Special:BookSources/978-0-470-59874-0"><bdi>978-0-470-59874-0</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/719886424">719886424</a><span class="reference-accessdate">. Retrieved <span class="nowrap">28 September</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8.+Fun+with+Fractions&rft.btitle=Technical+Math+For+Dummies&rft.place=Hoboken&rft.pages=120&rft.pub=Wiley+Publishing+Inc.&rft.date=2010&rft_id=info%3Aoclcnum%2F719886424&rft.isbn=978-0-470-59874-0&rft.aulast=Schoenborn&rft.aufirst=Barry&rft.au=Simkins%2C+Bradley&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftechnical-math-for-dummies_202007%2Fpage%2F120&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Fraction">"Fraction"</a>. Encyclopedia of Mathematics. 2012-04-06. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141021043927/http://www.encyclopediaofmath.org/index.php/Fraction">Archived</a> from the original on 2014-10-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fraction&rft.pub=Encyclopedia+of+Mathematics&rft.date=2012-04-06&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FFraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-galen-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-galen_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalen2004" class="citation journal cs1">Galen, Leslie Blackwell (March 2004). <a rel="nofollow" class="external text" href="http://www.integretechpub.com/research/papers/monthly238-242.pdf">"Putting Fractions in Their Place"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>111</b> (3): 238–242. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F4145131">10.2307/4145131</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4145131">4145131</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110713044149/http://www.integretechpub.com/research/papers/monthly238-242.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2011-07-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-01-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Putting+Fractions+in+Their+Place&rft.volume=111&rft.issue=3&rft.pages=238-242&rft.date=2004-03&rft_id=info%3Adoi%2F10.2307%2F4145131&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4145131%23id-name%3DJSTOR&rft.aulast=Galen&rft.aufirst=Leslie+Blackwell&rft_id=http%3A%2F%2Fwww.integretechpub.com%2Fresearch%2Fpapers%2Fmonthly238-242.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.allbusiness.com/glossaries/built-fraction/4955205-1.html">"built fraction"</a>. allbusiness.com glossary. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130526110042/http://www.allbusiness.com/glossaries/built-fraction/4955205-1.html">Archived</a> from the original on 2013-05-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-06-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=built+fraction&rft.pub=allbusiness.com+glossary&rft_id=http%3A%2F%2Fwww.allbusiness.com%2Fglossaries%2Fbuilt-fraction%2F4955205-1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.allbusiness.com/glossaries/piece-fraction/4949142-1.html">"piece fraction"</a>. allbusiness.com glossary. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130521071112/http://www.allbusiness.com/glossaries/piece-fraction/4949142-1.html">Archived</a> from the original on 2013-05-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-06-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=piece+fraction&rft.pub=allbusiness.com+glossary&rft_id=http%3A%2F%2Fwww.allbusiness.com%2Fglossaries%2Fpiece-fraction%2F4949142-1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-eves-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-eves_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1990" class="citation book cs1">Eves, Howard (1990). <i>An introduction to the history of mathematics</i> (6th ed.). Philadelphia: Saunders College Pub. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-029558-4" title="Special:BookSources/978-0-03-029558-4"><bdi>978-0-03-029558-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+history+of+mathematics&rft.place=Philadelphia&rft.edition=6th&rft.pub=Saunders+College+Pub.&rft.date=1990&rft.isbn=978-0-03-029558-4&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></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 id="CITEREFWinkler2004" class="citation book cs1"><a href="/wiki/Peter_Winkler" title="Peter Winkler">Winkler, Peter</a> (2004). "Uses of fuses". <i>Mathematical Puzzles: A Connoisseur's Collection</i>. A K Peters. pp. 2, 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-201-9" title="Special:BookSources/1-56881-201-9"><bdi>1-56881-201-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Uses+of+fuses&rft.btitle=Mathematical+Puzzles%3A+A+Connoisseur%27s+Collection&rft.pages=2%2C+6&rft.pub=A+K+Peters&rft.date=2004&rft.isbn=1-56881-201-9&rft.aulast=Winkler&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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="CITEREFCurtis1978" class="citation journal cs1">Curtis, Lorenzo J. (1978). "Concept of the exponential law prior to 1900". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>46</b> (9): 896–906. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978AmJPh..46..896C">1978AmJPh..46..896C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.11512">10.1119/1.11512</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Concept+of+the+exponential+law+prior+to+1900&rft.volume=46&rft.issue=9&rft.pages=896-906&rft.date=1978&rft_id=info%3Adoi%2F10.1119%2F1.11512&rft_id=info%3Abibcode%2F1978AmJPh..46..896C&rft.aulast=Curtis&rft.aufirst=Lorenzo+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-jeff-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-jeff_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-jeff_36-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-jeff_36-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-jeff_36-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller2014" class="citation web cs1">Miller, Jeff (22 December 2014). <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">"Earliest Uses of Various Mathematical Symbols"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160220073955/http://jeff560.tripod.com/mathsym.html">Archived</a> from the original on 20 February 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">15 February</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earliest+Uses+of+Various+Mathematical+Symbols&rft.date=2014-12-22&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fmathsym.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-filliozat-p152-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-filliozat-p152_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFilliozat2004" class="citation book cs1">Filliozat, Pierre-Sylvain (2004). "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature". In <a href="/wiki/Karine_Chemla" title="Karine Chemla">Chemla, Karine</a>; Cohen, Robert S.; Renn, Jürgen; et al. (eds.). <i>History of Science, History of Text</i>. Boston Series in the Philosophy of Science. Vol. 238. Dordrecht: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Netherlands</a>. p. 152. <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%2F1-4020-2321-9_7">10.1007/1-4020-2321-9_7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-2320-0" title="Special:BookSources/978-1-4020-2320-0"><bdi>978-1-4020-2320-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ancient+Sanskrit+Mathematics%3A+An+Oral+Tradition+and+a+Written+Literature&rft.btitle=History+of+Science%2C+History+of+Text&rft.place=Dordrecht&rft.series=Boston+Series+in+the+Philosophy+of+Science&rft.pages=152&rft.pub=Springer+Netherlands&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2F1-4020-2321-9_7&rft.isbn=978-1-4020-2320-0&rft.aulast=Filliozat&rft.aufirst=Pierre-Sylvain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1928" class="citation book cs1">Cajori, Florian (1928). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp"><i>A History of Mathematical Notations</i></a>. Vol. 1. La Salle, Illinois: Open Court Publishing Company. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp/page/n288">269</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140414021500/https://archive.org/details/historyofmathema031756mbp">Archived</a> from the original on 2014-04-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-08-30</span></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+Mathematical+Notations&rft.place=La+Salle%2C+Illinois&rft.pages=269&rft.pub=Open+Court+Publishing+Company&rft.date=1928&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema031756mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" 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"><a href="#CITEREFCajori1928">Cajori (1928)</a>, p. 89</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 class="citation book cs1"><i>A Source Book in Mathematics 1200–1800</i>. New Jersey: Princeton University Press. 1986. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02397-7" title="Special:BookSources/978-0-691-02397-7"><bdi>978-0-691-02397-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematics+1200%E2%80%931800&rft.place=New+Jersey&rft.pub=Princeton+University+Press&rft.date=1986&rft.isbn=978-0-691-02397-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī</i>. Wiesbaden: Steiner. 1951.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Rechenkunst+bei+%C4%9Eam%C5%A1%C4%ABd+b.+Mas%27%C5%ABd+al-K%C4%81%C5%A1%C4%AB&rft.place=Wiesbaden&rft.pub=Steiner&rft.date=1951&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerggren2007" class="citation book cs1">Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". <i>The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook</i>. Princeton University Press. p. 518. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+in+Medieval+Islam&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India%2C+and+Islam%3A+A+Sourcebook&rft.pages=518&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rft.aulast=Berggren&rft.aufirst=J.+Lennart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Uqlidisi.html">"MacTutor's al-Uqlidisi biography"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111115163359/http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Uqlidisi.html">Archived</a> 2011-11-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Retrieved 2011-11-22.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf">"Common Core State Standards for Mathematics"</a> <span class="cs1-format">(PDF)</span>. Common Core State Standards Initiative. 2010. p. 85. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131019052731/http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2013-10-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Common+Core+State+Standards+for+Mathematics&rft.pages=85&rft.pub=Common+Core+State+Standards+Initiative&rft.date=2010&rft_id=http%3A%2F%2Fwww.corestandards.org%2Fassets%2FCCSSI_Math%2520Standards.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></span> </li> </ol></div></div> <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 <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Fractions" class="extiw" title="commons:Category:Fractions">Fractions</a></span>.</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/denominator" class="extiw" title="wiktionary:denominator">denominator</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/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/numerator" class="extiw" title="wiktionary:numerator">numerator</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Fraction">"Fraction, arithmetical"</a>. <i>The Online Encyclopaedia of Mathematics</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fraction%2C+arithmetical&rft.btitle=The+Online+Encyclopaedia+of+Mathematics&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FFraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/EBchecked/topic/215508/fraction">"Fraction"</a>. <i>Encyclopædia Britannica</i>. 5 January 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fraction&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.date=2024-01-05&rft_id=https%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F215508%2Ffraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFraction" class="Z3988"></span></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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"></div><div role="navigation" class="navbox" aria-labelledby="Fractions_and_ratios" 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="4"><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:Fractions_and_ratios" title="Template:Fractions and ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fractions_and_ratios" title="Template talk:Fractions and ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fractions_and_ratios" title="Special:EditPage/Template:Fractions and ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fractions_and_ratios" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Fractions</a> and <a href="/wiki/Ratio" title="Ratio">ratios</a></div></th></tr><tr><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 2px 0 0;min-width: 60px"><div><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Unicode_0x0025.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/50px-Unicode_0x0025.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/75px-Unicode_0x0025.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/100px-Unicode_0x0025.svg.png 2x" data-file-width="16" data-file-height="16" /></a></span></div></td><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Division and ratio</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/Division_(mathematics)" title="Division (mathematics)">Dividend</a> ÷ <a href="/wiki/Divisor" title="Divisor">Divisor</a> = <a href="/wiki/Quotient" title="Quotient">Quotient</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File"><span><img alt="The ratio of width to height of standard-definition television." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/66px-Aspect-ratio-4x3.svg.png" decoding="async" width="66" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/99px-Aspect-ratio-4x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/131px-Aspect-ratio-4x3.svg.png 2x" data-file-width="139" data-file-height="106" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Fraction</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><span style="font-size:120%"><span class="sfrac">⁠<span class="tion"><span class="num">Numerator</span><span class="sr-only">/</span><span class="den">Denominator</span></span>⁠</span></span> = Quotient</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_fraction" title="Algebraic fraction">Algebraic</a></li> <li><a href="/wiki/Aspect_ratio" title="Aspect ratio">Aspect</a></li> <li><a href="/wiki/Binary_number" title="Binary number">Binary</a></li> <li><a href="/wiki/Continued_fraction" title="Continued fraction">Continued</a></li> <li><a href="/wiki/Decimal#Decimal_fractions" title="Decimal">Decimal</a></li> <li><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic</a></li> <li><a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden</a> <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver</a></li></ul></li> <li><a href="/wiki/Integer" title="Integer">Integer</a></li> <li><a href="/wiki/Irreducible_fraction" title="Irreducible fraction">Irreducible</a> <ul><li><a href="/wiki/Reduction_(mathematics)" title="Reduction (mathematics)">Reduction</a></li></ul></li> <li><a href="/wiki/Just_intonation" title="Just intonation">Just intonation</a></li> <li><a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">LCD</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">Musical interval</a></li> <li><a href="/wiki/Paper_size" title="Paper size">Paper size</a></li> <li><a href="/wiki/Percentage" title="Percentage">Percentage</a></li> <li><a href="/wiki/Unit_fraction" title="Unit fraction">Unit</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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></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/Q66055#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/4008387-1">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85051148">United States</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Fractions"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12063899h">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Fractions"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb12063899h">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00561041">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="zlomky"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph135439&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Fracciones"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX532372">Spain</a></span></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=987007548245905171">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=Fraction&oldid=1258329595">https://en.wikipedia.org/w/index.php?title=Fraction&oldid=1258329595</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Fractions_(mathematics)" title="Category:Fractions (mathematics)">Fractions (mathematics)</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: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:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_June_2023" title="Category:Articles needing additional references from June 2023">Articles needing additional references from June 2023</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: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_February_2016" title="Category:Articles with unsourced statements from February 2016">Articles with unsourced statements from February 2016</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:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category 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 19 November 2024, at 03:09<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=Fraction&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-694cf4987f-vsv6k","wgBackendResponseTime":236,"wgPageParseReport":{"limitreport":{"cputime":"1.182","walltime":"1.733","ppvisitednodes":{"value":7969,"limit":1000000},"postexpandincludesize":{"value":155678,"limit":2097152},"templateargumentsize":{"value":6896,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":16,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":229494,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1275.829 1 -total"," 29.67% 378.536 2 Template:Reflist"," 11.00% 140.354 22 Template:Cite_book"," 9.69% 123.648 7 Template:Cite_encyclopedia"," 8.90% 113.514 1 Template:Short_description"," 7.69% 98.173 1 Template:Harvtxt"," 7.44% 94.859 5 Template:Langx"," 6.20% 79.102 1 Template:Fractions_and_ratios"," 5.90% 75.253 1 Template:Navbox"," 5.85% 74.667 1 Template:More_citations_needed"]},"scribunto":{"limitreport-timeusage":{"value":"0.691","limit":"10.000"},"limitreport-memusage":{"value":19936602,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAmbrose_\u0026amp;_al.\"] = 1,\n [\"CITEREFBarlow1814\"] = 1,\n [\"CITEREFBerggren2007\"] = 1,\n [\"CITEREFBringhurst2002\"] = 1,\n [\"CITEREFCajori1928\"] = 1,\n [\"CITEREFCurtis1978\"] = 1,\n [\"CITEREFDavid_E._Smith1958\"] = 1,\n [\"CITEREFEasterday1982\"] = 1,\n [\"CITEREFEves1990\"] = 1,\n [\"CITEREFFilliozat2004\"] = 1,\n [\"CITEREFGalen2004\"] = 1,\n [\"CITEREFGardiner2016\"] = 1,\n [\"CITEREFGreer1986\"] = 1,\n [\"CITEREFJack_Williams2011\"] = 1,\n [\"CITEREFKelley2004\"] = 1,\n [\"CITEREFLaurel_BrennerPeterson2004\"] = 1,\n [\"CITEREFLeeMessner2000\"] = 1,\n [\"CITEREFMiller2014\"] = 1,\n [\"CITEREFPerryPerry1981\"] = 1,\n [\"CITEREFRecord1654\"] = 1,\n [\"CITEREFSchoenbornSimkins2010\"] = 1,\n [\"CITEREFSchwartzman1994\"] = 1,\n [\"CITEREFTrotter1853\"] = 1,\n [\"CITEREFWeisstein2003\"] = 1,\n [\"CITEREFWingard-Nelson2014\"] = 1,\n [\"CITEREFWinkler2004\"] = 1,\n [\"CITEREFWu2011\"] = 1,\n}\ntemplate_list = table#1 {\n [\"3\"] = 2,\n [\"Angle brackets\"] = 4,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 5,\n [\"Citation needed\"] = 2,\n [\"Cite book\"] = 22,\n [\"Cite encyclopedia\"] = 7,\n [\"Cite journal\"] = 4,\n [\"Cite web\"] = 9,\n [\"Classification of numbers\"] = 1,\n [\"Commons category\"] = 1,\n [\"DEFAULTSORT:Fraction (Mathematics)\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Floruit\"] = 1,\n [\"Frac\"] = 1,\n [\"Fraction\"] = 1,\n [\"Fractions and ratios\"] = 1,\n [\"Further\"] = 1,\n [\"Harvid\"] = 1,\n [\"Harvp\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"Lang\"] = 1,\n [\"Langx\"] = 5,\n [\"Main\"] = 2,\n [\"Math\"] = 1,\n [\"More citations needed\"] = 1,\n [\"Mvar\"] = 3,\n [\"Nobr\"] = 3,\n [\"Not a typo\"] = 2,\n [\"Nowrap\"] = 6,\n [\"Other uses\"] = 1,\n [\"Overline\"] = 15,\n [\"Pgs\"] = 1,\n [\"Pp-semi-indef\"] = 1,\n [\"ProQuest\"] = 2,\n [\"Reflist\"] = 2,\n [\"Refn\"] = 2,\n [\"See also\"] = 3,\n [\"Sfrac\"] = 69,\n [\"Short description\"] = 1,\n [\"Slink\"] = 2,\n [\"Snd\"] = 1,\n [\"Tmath\"] = 14,\n [\"Val\"] = 16,\n [\"Webarchive\"] = 2,\n [\"Wiktionary\"] = 2,\n [\"\\\\color{RedOrange\"] = 2,\n [\"\\\\color{RoyalBlue\"] = 2,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-6bv29","timestamp":"20241125142844","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Fraction","url":"https:\/\/en.wikipedia.org\/wiki\/Fraction","sameAs":"http:\/\/www.wikidata.org\/entity\/Q66055","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q66055","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":"2005-04-07T14:36:32Z","dateModified":"2024-11-19T03:09:54Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b9\/Cake_quarters.svg","headline":"mathematical representation of a portion of a whole"}</script> </body> </html>

CINXE.COM

Fraction - Wikipedia