CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Parity (mathematics) - 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":"f2f742d7-5722-4b52-9fa1-50851993ab63","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Parity_(mathematics)","wgTitle":"Parity (mathematics)","wgCurRevisionId":1259783707,"wgRevisionId":1259783707,"wgArticleId":143135,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Parity (mathematics)","Elementary arithmetic"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Parity_(mathematics)","wgRelevantArticleId":143135,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false, "wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q230967","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","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%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/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="776"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="517"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="414"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Parity (mathematics) - 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/Parity_(mathematics)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Parity_(mathematics)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Parity_(mathematics)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Parity_mathematics rootpage-Parity_mathematics 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=Parity+%28mathematics%29" 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=Parity+%28mathematics%29" 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=Parity+%28mathematics%29" 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=Parity+%28mathematics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Addition_and_subtraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition_and_subtraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Addition and subtraction</span> </div> </a> <ul id="toc-Addition_and_subtraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication" 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">2.2</span> <span>Multiplication</span> </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> </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">2.3</span> <span>Division</span> </div> </a> <ul id="toc-Division-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Higher mathematics</span> </div> </a> <button aria-controls="toc-Higher_mathematics-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 Higher mathematics subsection</span> </button> <ul id="toc-Higher_mathematics-sublist" class="vector-toc-list"> <li id="toc-Higher_dimensions_and_more_general_classes_of_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_dimensions_and_more_general_classes_of_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Higher dimensions and more general classes of numbers</span> </div> </a> <ul id="toc-Higher_dimensions_and_more_general_classes_of_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Number theory</span> </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Group theory</span> </div> </a> <ul id="toc-Group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Analysis</span> </div> </a> <ul id="toc-Analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_game_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_game_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Combinatorial game theory</span> </div> </a> <ul id="toc-Combinatorial_game_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Additional_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Additional applications</span> </div> </a> <ul id="toc-Additional_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-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">Parity (mathematics)</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 68 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-68" 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">68 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/Pariteit" title="Pariteit – Afrikaans" lang="af" hreflang="af" data-title="Pariteit" 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/%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF_%D8%B2%D9%88%D8%AC%D9%8A%D8%A9_%D9%88%D9%81%D8%B1%D8%AF%D9%8A%D8%A9" title="أعداد زوجية وفردية – Arabic" lang="ar" hreflang="ar" data-title="أعداد زوجية وفردية" 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%AF%E0%A7%81%E0%A6%97%E0%A7%8D%E0%A6%AE_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="যুগ্ম সংখ্যা – Assamese" lang="as" hreflang="as" data-title="যুগ্ম সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99k_v%C9%99_c%C3%BCt_%C9%99d%C9%99dl%C9%99r" title="Tək və cüt ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Tək və cüt ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AA%DA%A9_%D9%88_%D8%AC%D9%88%D8%AA_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="تک و جوت ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="تک و جوت ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A7%8B%E0%A6%A1%E0%A6%BC_%E0%A6%93_%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8B%E0%A6%A1%E0%A6%BC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জোড় ও বিজোড় সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="জোড় ও বিজোড় সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Khia-s%C3%B2%CD%98_kap_ng%C3%B3%CD%98-s%C3%B2%CD%98" title="Khia-sò͘ kap ngó͘-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Khia-sò͘ kap ngó͘-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%99%D0%BE%D0%BF%D0%BB%D0%BE%D2%A1" title="Йоплоҡ – Bashkir" lang="ba" hreflang="ba" data-title="Йоплоҡ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%BD%D0%BE%D1%81%D1%82" title="Четност – Bulgarian" lang="bg" hreflang="bg" data-title="Четност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_parell" title="Nombre parell – Catalan" lang="ca" hreflang="ca" data-title="Nombre parell" 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%AB%D1%82-%D1%82%C4%95%D0%BA%D0%B5%D0%BB" title="Ыт-тĕкел – Chuvash" lang="cv" hreflang="cv" data-title="Ыт-тĕкел" 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/Sud%C3%A1_a_lich%C3%A1_%C4%8D%C3%ADsla" title="Sudá a lichá čísla – Czech" lang="cs" hreflang="cs" data-title="Sudá a lichá čísla" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Paridad_(matematica)" title="Paridad (matematica) – Chavacano" lang="cbk" hreflang="cbk" data-title="Paridad (matematica)" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Paredd_(mathemateg)" title="Paredd (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Paredd (mathemateg)" 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/Paritet_(talteori)" title="Paritet (talteori) – Danish" lang="da" hreflang="da" data-title="Paritet (talteori)" 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/Parit%C3%A4t_(Mathematik)" title="Parität (Mathematik) – German" lang="de" hreflang="de" data-title="Parität (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CF%81%CF%84%CE%B9%CE%BF%CE%B9_%CE%BA%CE%B1%CE%B9_%CF%80%CE%B5%CF%81%CE%B9%CF%84%CF%84%CE%BF%CE%AF_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CE%AF" title="Άρτιοι και περιττοί αριθμοί – Greek" lang="el" hreflang="el" data-title="Άρτιοι και περιττοί αριθμοί" 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/N%C3%BAmeros_pares_e_impares" title="Números pares e impares – Spanish" lang="es" hreflang="es" data-title="Números pares e impares" 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/Pareco_de_nombroj" title="Pareco de nombroj – Esperanto" lang="eo" hreflang="eo" data-title="Pareco de nombroj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%B2%D9%88%D8%AC_%D9%88_%D9%81%D8%B1%D8%AF" title="اعداد زوج و فرد – Persian" lang="fa" hreflang="fa" data-title="اعداد زوج و فرد" 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/Parit%C3%A9_(arithm%C3%A9tique)" title="Parité (arithmétique) – French" lang="fr" hreflang="fr" data-title="Parité (arithmétique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Paridade_(matem%C3%A1ticas)" title="Paridade (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Paridade (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%80%EC%88%98%EC%99%80_%EC%A7%9D%EC%88%98" title="홀수와 짝수 – Korean" lang="ko" hreflang="ko" data-title="홀수와 짝수" 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%B6%D5%B8%D6%82%D5%B5%D5%A3_%D6%87_%D5%AF%D5%A5%D5%B6%D5%BF_%D5%A9%D5%BE%D5%A5%D6%80" title="Զույգ և կենտ թվեր – Armenian" lang="hy" hreflang="hy" data-title="Զույգ և կենտ թվեր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE_%E0%A4%94%E0%A4%B0_%E0%A4%B5%E0%A4%BF%E0%A4%B7%E0%A4%AE_%E0%A4%85%E0%A4%82%E0%A4%95" title="सम और विषम अंक – Hindi" lang="hi" hreflang="hi" data-title="सम और विषम अंक" 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/Parnost_broja" title="Parnost broja – Croatian" lang="hr" hreflang="hr" data-title="Parnost broja" 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/Pareso" title="Pareso – Ido" lang="io" hreflang="io" data-title="Pareso" 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/Paritas_(matematika)" title="Paritas (matematika) – Indonesian" lang="id" hreflang="id" data-title="Paritas (matematika)" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_pari_e_dispari" title="Numeri pari e dispari – Italian" lang="it" hreflang="it" data-title="Numeri pari e dispari" 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%96%D7%95%D7%92%D7%99%D7%95%D7%AA_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="זוגיות (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="זוגיות (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%A3%E1%83%AC%E1%83%98_%E1%83%93%E1%83%90_%E1%83%99%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%94%E1%83%91%E1%83%98" title="ლუწი და კენტი რიცხვები – Georgian" lang="ka" hreflang="ka" data-title="ლუწი და კენტი რიცხვები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D2%B1%D0%BF_%D1%81%D0%B0%D0%BD" title="Жұп сан – Kazakh" lang="kk" hreflang="kk" data-title="Жұп сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Usawa_(hisabati)" title="Usawa (hisabati) – Swahili" lang="sw" hreflang="sw" data-title="Usawa (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Parit%C3%A9it_(Mathematik)" title="Paritéit (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Paritéit (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Lyginiai_ir_nelyginiai_skai%C4%8Diai" title="Lyginiai ir nelyginiai skaičiai – Lithuanian" lang="lt" hreflang="lt" data-title="Lyginiai ir nelyginiai skaičiai" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/P%C3%A1ros_%C3%A9s_p%C3%A1ratlan_sz%C3%A1mok" title="Páros és páratlan számok – Hungarian" lang="hu" hreflang="hu" data-title="Páros és páratlan számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Ankasa_sy_tsiankasa" title="Ankasa sy tsiankasa – Malagasy" lang="mg" hreflang="mg" data-title="Ankasa sy tsiankasa" 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%87%E0%B4%B0%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="ഇരട്ടസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="ഇരട്ടസംഖ്യ" 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/Parit%C3%A0_(matematika)" title="Parità (matematika) – Maltese" lang="mt" hreflang="mt" data-title="Parità (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/Nombor_genap_dan_ganjil" title="Nombor genap dan ganjil – Malay" lang="ms" hreflang="ms" data-title="Nombor genap dan ganjil" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pariteit_(wiskunde)" title="Pariteit (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Pariteit (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%81%B6%E5%A5%87%E6%80%A7" title="偶奇性 – Japanese" lang="ja" hreflang="ja" data-title="偶奇性" 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/Pariteet" title="Pariteet – Northern Frisian" lang="frr" hreflang="frr" data-title="Pariteet" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Oddetal_og_partal" title="Oddetal og partal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Oddetal og partal" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Juft_va_toq_sonlar" title="Juft va toq sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Juft va toq sonlar" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_cobi" title="Nùmer cobi – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer cobi" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Evene_un_unevene_Tallen" title="Evene un unevene Tallen – Low German" lang="nds" hreflang="nds" data-title="Evene un unevene Tallen" 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/Parzysto%C5%9B%C4%87_liczb" title="Parzystość liczb – Polish" lang="pl" hreflang="pl" data-title="Parzystość liczb" 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/Paridade" title="Paridade – Portuguese" lang="pt" hreflang="pt" data-title="Paridade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D1%91%D1%82%D0%BD%D1%8B%D0%B5_%D0%B8_%D0%BD%D0%B5%D1%87%D1%91%D1%82%D0%BD%D1%8B%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Чётные и нечётные числа – Russian" lang="ru" hreflang="ru" data-title="Чётные и нечётные числа" 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/%C3%87ift%C3%ABsia_(matematik%C3%AB)" title="Çiftësia (matematikë) – Albanian" lang="sq" hreflang="sq" data-title="Çiftësia (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/P%C3%A1rne_%C4%8D%C3%ADslo" title="Párne číslo – Slovak" lang="sk" hreflang="sk" data-title="Párne číslo" 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/Soda_in_liha_%C5%A1tevila" title="Soda in liha števila – Slovenian" lang="sl" hreflang="sl" data-title="Soda in liha števila" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%BD%D0%B8_%D0%B8_%D0%BD%D0%B5%D0%BF%D0%B0%D1%80%D0%BD%D0%B8_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B8" title="Парни и непарни бројеви – Serbian" lang="sr" hreflang="sr" data-title="Парни и непарни бројеви" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pariteetti_(matematiikka)" title="Pariteetti (matematiikka) – Finnish" lang="fi" hreflang="fi" data-title="Pariteetti (matematiikka)" 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/J%C3%A4mna_och_udda_tal" title="Jämna och udda tal – Swedish" lang="sv" hreflang="sv" data-title="Jämna och udda tal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kapantayan_(matematika)" title="Kapantayan (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Kapantayan (matematika)" 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%A8%E0%AE%BF%E0%AE%95%E0%AE%B0%E0%AE%BF_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="நிகரி (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="நிகரி (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A0%E0%B8%B2%E0%B8%A7%E0%B8%B0%E0%B8%84%E0%B8%B9%E0%B9%88%E0%B8%AB%E0%B8%A3%E0%B8%B7%E0%B8%AD%E0%B8%84%E0%B8%B5%E0%B9%88_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ภาวะคู่หรือคี่ (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="ภาวะคู่หรือคี่ (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D2%B3%D0%BE%D0%B8_%D2%B7%D1%83%D1%84%D1%82_%D0%B2%D0%B0_%D1%82%D0%BE%D2%9B" title="Ададҳои ҷуфт ва тоқ – Tajik" lang="tg" hreflang="tg" data-title="Ададҳои ҷуфт ва тоқ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Parite_(matematik)" title="Parite (matematik) – Turkish" lang="tr" hreflang="tr" data-title="Parite (matematik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%BD%D1%96%D1%81%D1%82%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Парність (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Парність (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADnh_ch%E1%BA%B5n_l%E1%BA%BB" title="Tính chẵn lẻ – Vietnamese" lang="vi" hreflang="vi" data-title="Tính chẵn lẻ" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A5%87%E5%81%B6%E6%80%A7%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="奇偶性（数学） – Wu" lang="wuu" hreflang="wuu" data-title="奇偶性（数学）" 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%92%D7%A8%D7%90%D7%93_(%D7%A0%D7%95%D7%9E%D7%A2%D7%A8)" title="גראד (נומער) – Yiddish" lang="yi" hreflang="yi" data-title="גראד (נומער)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_ad%E1%BB%8D%CC%81gba_%C3%A0ti_a%E1%B9%A3%E1%BA%B9%CC%81k%C3%B9" title="Nọ́mbà adọ́gba àti aṣẹ́kù – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà adọ́gba àti aṣẹ́kù" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%96%AE%E9%9B%99%E6%95%B8" title="單雙數 – Cantonese" lang="yue" hreflang="yue" data-title="單雙數" 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%A5%87%E5%81%B6%E6%80%A7_(%E6%95%B0%E5%AD%A6)" title="奇偶性 (数学) – Chinese" lang="zh" hreflang="zh" data-title="奇偶性 (数学)" 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/Q230967#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/Parity_(mathematics)" 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:Parity_(mathematics)" 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/Parity_(mathematics)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Parity_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Parity_(mathematics)&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/Parity_(mathematics)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Parity_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Parity_(mathematics)&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/Parity_(mathematics)" 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/Parity_(mathematics)" 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=Parity_(mathematics)&oldid=1259783707" 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=Parity_(mathematics)&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=Parity_%28mathematics%29&id=1259783707&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%2FParity_%28mathematics%29"><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%2FParity_%28mathematics%29"><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=Parity_%28mathematics%29&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=Parity_(mathematics)&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:Parity_(mathematics)" 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/Odds_and_Evens" 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/Q230967" 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> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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/Parity_(disambiguation)" class="mw-redirect mw-disambig" title="Parity (disambiguation)">Parity (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Odd number" redirects here. For the 1962 Argentine film, see <a href="/wiki/Odd_Number_(film)" title="Odd Number (film)"><i>Odd Number</i> (film)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of being an even or odd number</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Parity_of_5_and_6_Cuisenaire_rods.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png/275px-Parity_of_5_and_6_Cuisenaire_rods.png" decoding="async" width="275" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png/413px-Parity_of_5_and_6_Cuisenaire_rods.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png/550px-Parity_of_5_and_6_Cuisenaire_rods.png 2x" data-file-width="600" data-file-height="388" /></a><figcaption><a href="/wiki/Cuisenaire_rods" title="Cuisenaire rods">Cuisenaire rods</a>: 5 (yellow) <i>cannot</i> be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) <i>can</i> be evenly divided in 2 by 3 (lime green).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>parity</b> is the <a href="/wiki/Property_(mathematics)" title="Property (mathematics)">property</a> of an <a href="/wiki/Integer" title="Integer">integer</a> of whether it is <b>even</b> or <b>odd</b>. An integer is even if it is <a href="/wiki/Divisible" class="mw-redirect" title="Divisible">divisible</a> by 2, and odd if it is not.<sup id="cite_ref-rod_1-0" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, −4, 0, and 82 are even numbers, while −3, 5, 7, and 21 are odd numbers. </p><p>The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. </p><p>Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the <a href="/wiki/Parity_of_zero" title="Parity of zero">parity of zero</a> is even.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the <a href="/wiki/Decimal" title="Decimal">decimal</a> <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a> is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An even number is an integer of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4d12db9542ff955cca683305e0a0bcd0a1d217" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.802ex; height:2.176ex;" alt="{\displaystyle x=2k}"></span> where <i>k</i> is an integer;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> an odd number is an integer of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2k+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2k+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7399f628b032e84a261d153e39983d1aa6c68b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.452ex; height:2.343ex;" alt="{\displaystyle x=2k+1.}"></span> </p><p>An equivalent definition is that an even number is <a href="/wiki/Divisible" class="mw-redirect" title="Divisible">divisible</a> by 2: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\ |\ x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ |\ x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d114d24ad4aeddf64778c66eb0012f4ba9020e99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.3ex; height:2.843ex;" alt="{\displaystyle 2\ |\ x}"></span> and an odd number is not: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\not |\ x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\not |\ x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2639dc61412b20918ee0cd758bf6fd7d1d75a2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.01ex; height:2.843ex;" alt="{\displaystyle 2\not |\ x}"></span> </p><p>The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> of even and odd numbers can be defined as following:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2k:k\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mi>k</mi> <mo>:</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2k:k\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb356b6eeb8b46650a6d3abb71a72474b979c6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.238ex; height:2.843ex;" alt="{\displaystyle \{2k:k\in \mathbb {Z} \}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2k+1:k\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>:</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2k+1:k\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc766fa2c90d84f19d35cb17b3f4ce4627b4b41d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.241ex; height:2.843ex;" alt="{\displaystyle \{2k+1:k\in \mathbb {Z} \}}"></span> </p><p>The set of <i>even</i> numbers is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> 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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> and the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</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} /2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }"></span> is the <a href="/wiki/Field_with_two_elements" class="mw-redirect" title="Field with two elements">field with two elements</a>. Parity can then be defined as the unique <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> 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 \mathbb {Z} /2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }"></span> where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following laws can be verified using the properties of <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>. They are a special case of rules in <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. </p> <div class="mw-heading mw-heading3"><h3 id="Addition_and_subtraction">Addition and subtraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=3" title="Edit section: Addition and subtraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>even ± even = even;<sup id="cite_ref-rod_1-1" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>even ± odd = odd;<sup id="cite_ref-rod_1-2" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>odd ± odd = even;<sup id="cite_ref-rod_1-3" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=4" title="Edit section: Multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>even × even = even;<sup id="cite_ref-rod_1-4" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>even × odd = even;<sup id="cite_ref-rod_1-5" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>odd × odd = odd;<sup id="cite_ref-rod_1-6" class="reference"><a href="#cite_note-rod-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <p>By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the <a href="/wiki/GF(2)" title="GF(2)">field with two elements</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Division">Division</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=5" title="Edit section: Division"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even <i>nor</i> odd, since the concepts of even and odd apply only to integers. But when the <a href="/wiki/Quotient" title="Quotient">quotient</a> is an integer, it will be even <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">dividend</a> has more <a href="/wiki/Integer_factorization" title="Integer factorization">factors of two</a> than the divisor.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ancient Greeks considered 1, the <a href="/wiki/Monad_(philosophy)" title="Monad (philosophy)">monad</a>, to be neither fully odd nor fully even.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Some of this sentiment survived into the 19th century: <a href="/wiki/Friedrich_Fr%C3%B6bel" title="Friedrich Fröbel">Friedrich Wilhelm August Fröbel</a>'s 1826 <i>The Education of Man</i> instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches the philosophical afterthought, </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form, the right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Higher_mathematics">Higher mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=7" title="Edit section: Higher mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Higher_dimensions_and_more_general_classes_of_numbers">Higher dimensions and more general classes of numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=8" title="Edit section: Higher dimensions and more general classes of numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="thumb noviewer tright"><div class="center" style="line-height:130%;margin:0 auto;max-width:254px"> </div><div class="thumbinner" style="width:246px"><table cellpadding="0" cellspacing="0" style="font-size:88%;border:1px #c8ccd1 solid;padding:0;margin:auto"><tbody><tr style="vertical-align:middle"><td style="vertical-align:inherit;padding:0"></td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">a</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">b</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">c</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">d</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">e</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">f</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">g</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">h</td><td style="vertical-align:inherit;padding:0"></td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;width:18px;height:26px">8</td><td colspan="8" rowspan="8" style="padding:0;vertical-align:inherit"><div class="chess-board notheme" style="position:relative"><span class="notpageimage" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Chessboard480.svg/208px-Chessboard480.svg.png" decoding="async" width="208" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Chessboard480.svg/312px-Chessboard480.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Chessboard480.svg/416px-Chessboard480.svg.png 2x" data-file-width="480" data-file-height="480" /></span></span><div style="position:absolute;z-index:3;top:0px;left:52px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="c8 black cross"><img alt="c8 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:0px;left:104px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="e8 black cross"><img alt="e8 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:26px;left:26px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="b7 black cross"><img alt="b7 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:26px;left:130px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="f7 black cross"><img alt="f7 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:52px;left:78px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="d6 black knight"><img alt="d6 black knight" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Chess_ndt45.svg/26px-Chess_ndt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Chess_ndt45.svg/39px-Chess_ndt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Chess_ndt45.svg/52px-Chess_ndt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:78px;left:26px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="b5 black cross"><img alt="b5 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:78px;left:130px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="f5 black cross"><img alt="f5 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:104px;left:52px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="c4 black cross"><img alt="c4 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:104px;left:104px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="e4 black cross"><img alt="e4 black cross" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/26px-Chess_xxt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/39px-Chess_xxt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chess_xxt45.svg/52px-Chess_xxt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:182px;left:52px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="c1 white bishop"><img alt="c1 white bishop" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/26px-Chess_blt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/39px-Chess_blt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/52px-Chess_blt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div><div style="position:absolute;z-index:3;top:182px;left:130px;width:26px;height:26px"><span class="mw-valign-top notpageimage" typeof="mw:File"><span title="f1 white bishop"><img alt="f1 white bishop" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/26px-Chess_blt45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/39px-Chess_blt45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Chess_blt45.svg/52px-Chess_blt45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></div></div></td><td style="padding:0;vertical-align:inherit;text-align:center;width:18px;height:26px">8</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">7</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">7</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">6</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">6</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">5</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">5</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">4</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">4</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">3</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">3</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">2</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">2</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">1</td><td style="padding:0;vertical-align:inherit;text-align:center;height:26px">1</td></tr><tr style="vertical-align:middle"><td style="vertical-align:inherit;padding:0"></td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">a</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">b</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">c</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">d</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">e</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">f</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">g</td><td style="padding:0;vertical-align:inherit;text-align:center;height:18px;width:26px">h</td><td style="vertical-align:inherit;padding:0"></td></tr></tbody></table><div class="thumbcaption"> Each of the white <a href="/wiki/Bishop_(chess)" title="Bishop (chess)">bishops</a> is confined to squares of the same parity; the black <a href="/wiki/Knight_(chess)" title="Knight (chess)">knight</a> can only jump to squares of alternating parity. </div></div></div> <p>Integer coordinates of points in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a> of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">face-centered cubic lattice</a> and its higher-dimensional generalizations (the <i>D<sub>n</sub></i> <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattices</a>) consist of all of the integer points whose coordinates have an even sum.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This feature also manifests itself in <a href="/wiki/Chess" title="Chess">chess</a>, where the parity of a square is indicated by its color: <a href="/wiki/Bishop_(chess)" title="Bishop (chess)">bishops</a> are constrained to moving between squares of the same parity, whereas <a href="/wiki/Knight_(chess)" title="Knight (chess)">knights</a> alternate parity between moves.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This form of parity was famously used to solve the <a href="/wiki/Mutilated_chessboard_problem" title="Mutilated chessboard problem">mutilated chessboard problem</a>: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Even_and_odd_ordinals" title="Even and odd ordinals">parity of an ordinal number</a> may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Let <i>R</i> be a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> and let <i>I</i> be an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of <i>R</i> whose <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> is 2. Elements of the <a href="/wiki/Coset" title="Coset">coset</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 0+I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ca26843d7c2d02e4137b17c0bfda329d4af3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.175ex; height:2.343ex;" alt="{\displaystyle 0+I}"></span> may be called <b>even</b>, while elements of the coset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053e02c01151ce5f4485e583554906a3fee729a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.175ex; height:2.343ex;" alt="{\displaystyle 1+I}"></span> may be called <b>odd</b>. As an example, let <span class="texhtml"><i>R</i> = <b>Z</b><sub>(2)</sub></span> be the <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> of <b>Z</b> at the <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> (2). Then an element of <i>R</i> is even or odd if and only if its numerator is so in <b>Z</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=9" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The even numbers form an <a href="/wiki/Ring_ideal" class="mw-redirect" title="Ring ideal">ideal</a> in the <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a> of integers,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> but the odd numbers do not—this is clear from the fact that the <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a> element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. </p><p>All <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> are odd, with one exception: the prime number 2.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> All known <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a> are even; it is unknown whether any odd perfect numbers exist.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern <a href="/wiki/Computer" title="Computer">computer</a> calculations have shown this conjecture to be true for integers up to at least 4 × 10<sup>18</sup>, but still no general <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> has been found.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Group_theory">Group theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=10" title="Edit section: Group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Rubiks_revenge_solved.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Rubiks_revenge_solved.jpg/220px-Rubiks_revenge_solved.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Rubiks_revenge_solved.jpg/330px-Rubiks_revenge_solved.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Rubiks_revenge_solved.jpg/440px-Rubiks_revenge_solved.jpg 2x" data-file-width="600" data-file-height="600" /></a><figcaption><a href="/wiki/Rubik%27s_Revenge" title="Rubik's Revenge">Rubik's Revenge</a> in solved state</figcaption></figure> <p>The <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">parity of a permutation</a> (as defined in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>) is the parity of the number of <a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">transpositions</a> into which the permutation can be decomposed.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In <a href="/wiki/Rubik%27s_Cube" title="Rubik's Cube">Rubik's Cube</a>, <a href="/wiki/Megaminx" title="Megaminx">Megaminx</a>, and other twisting puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the <a href="/wiki/Configuration_space_(mathematics)" title="Configuration space (mathematics)">configuration space</a> of these puzzles.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Feit%E2%80%93Thompson_theorem" title="Feit–Thompson theorem">Feit–Thompson theorem</a> states that a <a href="/wiki/Finite_group" title="Finite group">finite group</a> is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Analysis">Analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=11" title="Edit section: Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">parity of a function</a> describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case <i>f</i>(<i>x</i>) = 0, to be both odd and even.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_game_theory">Combinatorial game theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=12" title="Edit section: Combinatorial game theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a>, an <i>evil number</i> is a number that has an even number of 1's in its <a href="/wiki/Binary_representation" class="mw-redirect" title="Binary representation">binary representation</a>, and an <i>odious number</i> is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game <a href="/wiki/Kayles" title="Kayles">Kayles</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Parity_function" title="Parity function">parity function</a> maps a number to the number of 1's in its binary representation, <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo 2</a>, so its value is zero for evil numbers and one for odious numbers. The <a href="/wiki/Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a>, an infinite sequence of 0's and 1's, has a 0 in position <i>i</i> when <i>i</i> is evil, and a 1 in that position when <i>i</i> is odious.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Additional_applications">Additional applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=13" title="Edit section: Additional applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Information_theory" title="Information theory">information theory</a>, a <a href="/wiki/Parity_bit" title="Parity bit">parity bit</a> appended to a binary number provides the simplest form of <a href="/wiki/Error_detecting_code" class="mw-redirect" title="Error detecting code">error detecting code</a>. If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing the number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Wind_instrument" title="Wind instrument">wind instruments</a> with a cylindrical bore and in effect closed at one end, such as the <a href="/wiki/Clarinet" title="Clarinet">clarinet</a> at the mouthpiece, the <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> produced are odd multiples of the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>. (With cylindrical pipes open at both ends, used for example in some <a href="/wiki/Organ_stop" title="Organ stop">organ stops</a> such as the <a href="/wiki/Flue_pipe#Diapasons" title="Flue pipe">open diapason</a>, the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series (music)</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>In some countries, <a href="/wiki/House_numbering" title="House numbering">house numberings</a> are chosen so that the houses on one side of a street have even numbers and the houses on the other side have odd numbers.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Similarly, among <a href="/wiki/United_States_numbered_highways" class="mw-redirect" title="United States numbered highways">United States numbered highways</a>, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Among airline <a href="/wiki/Flight_number" title="Flight number">flight numbers</a>, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Half-integer" title="Half-integer">Half-integer</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parity_(mathematics)&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-rod-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-rod_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rod_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-rod_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-rod_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-rod_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-rod_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-rod_1-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVijayaRodriguez" class="citation cs2">Vijaya, A.V.; Rodriguez, Dora, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9ZN9LuHb0tQC&pg=PA20"><i>Figuring Out Mathematics</i></a>, Pearson Education India, pp. 20–21, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788131703571" title="Special:BookSources/9788131703571"><bdi>9788131703571</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Figuring+Out+Mathematics&rft.pages=20-21&rft.pub=Pearson+Education+India&rft.isbn=9788131703571&rft.aulast=Vijaya&rft.aufirst=A.V.&rft.au=Rodriguez%2C+Dora&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9ZN9LuHb0tQC%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBóna2011" class="citation cs2">Bóna, Miklós (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TzJ2L9ZmlQUC&pg=PA178"><i>A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory</i></a>, World Scientific, p. 178, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789814335232" title="Special:BookSources/9789814335232"><bdi>9789814335232</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+Walk+Through+Combinatorics%3A+An+Introduction+to+Enumeration+and+Graph+Theory&rft.pages=178&rft.pub=World+Scientific&rft.date=2011&rft.isbn=9789814335232&rft.aulast=B%C3%B3na&rft.aufirst=Mikl%C3%B3s&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTzJ2L9ZmlQUC%26pg%3DPA178&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFOwen1992" class="citation cs2">Owen, Ruth L. (1992), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150317173427/http://www.pentagon.kappamuepsilon.org/pentagon/Vol_51_Num_2_Spring_1992.pdf">"Divisibility in bases"</a> <span class="cs1-format">(PDF)</span>, <i>The Pentagon: A Mathematics Magazine for Students</i>, <b>51</b> (2): 17–20, archived from <a rel="nofollow" class="external text" href="http://www.pentagon.kappamuepsilon.org/pentagon/Vol_51_Num_2_Spring_1992.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2015-03-17</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Pentagon%3A+A+Mathematics+Magazine+for+Students&rft.atitle=Divisibility+in+bases&rft.volume=51&rft.issue=2&rft.pages=17-20&rft.date=1992&rft.aulast=Owen&rft.aufirst=Ruth+L.&rft_id=http%3A%2F%2Fwww.pentagon.kappamuepsilon.org%2Fpentagon%2FVol_51_Num_2_Spring_1992.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBassarear2010" class="citation cs2">Bassarear, Tom (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RitXafH4_8EC&pg=PA198"><i>Mathematics for Elementary School Teachers</i></a>, Cengage Learning, p. 198, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780840054630" title="Special:BookSources/9780840054630"><bdi>9780840054630</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Elementary+School+Teachers&rft.pages=198&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=9780840054630&rft.aulast=Bassarear&rft.aufirst=Tom&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRitXafH4_8EC%26pg%3DPA198&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSidebotham2003" class="citation cs2">Sidebotham, Thomas H. (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VsAZa5PWLz8C&pg=PA181"><i>The A to Z of Mathematics: A Basic Guide</i></a>, John Wiley & Sons, p. 181, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471461630" title="Special:BookSources/9780471461630"><bdi>9780471461630</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+A+to+Z+of+Mathematics%3A+A+Basic+Guide&rft.pages=181&rft.pub=John+Wiley+%26+Sons&rft.date=2003&rft.isbn=9780471461630&rft.aulast=Sidebotham&rft.aufirst=Thomas+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVsAZa5PWLz8C%26pg%3DPA181&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPólyaTarjanWoods2009" class="citation cs2"><a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya, George</a>; <a href="/wiki/Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a>; Woods, Donald R. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y6KmsI0Icp0C&pg=PA21"><i>Notes on Introductory Combinatorics</i></a>, Springer, pp. 21–22, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780817649524" title="Special:BookSources/9780817649524"><bdi>9780817649524</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+Introductory+Combinatorics&rft.pages=21-22&rft.pub=Springer&rft.date=2009&rft.isbn=9780817649524&rft.aulast=P%C3%B3lya&rft.aufirst=George&rft.au=Tarjan%2C+Robert+E.&rft.au=Woods%2C+Donald+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dy6KmsI0Icp0C%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTankha2006" class="citation cs2">Tankha (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=88PFcpKjupAC&pg=PT126"><i>Ancient Greek Philosophy: Thales to Gorgias</i></a>, Pearson Education India, p. 126, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788177589399" title="Special:BookSources/9788177589399"><bdi>9788177589399</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ancient+Greek+Philosophy%3A+Thales+to+Gorgias&rft.pages=126&rft.pub=Pearson+Education+India&rft.date=2006&rft.isbn=9788177589399&rft.au=Tankha&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D88PFcpKjupAC%26pg%3DPT126&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFroebel1885" class="citation cs2">Froebel, Friedrich (1885), <a rel="nofollow" class="external text" href="https://archive.org/details/educationofman00froe"><i>The Education of Man</i></a>, translated by Jarvis, Josephine, New York: A Lovell & Company, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/educationofman00froe/page/240">240</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Education+of+Man&rft.place=New+York&rft.pages=240&rft.pub=A+Lovell+%26+Company&rft.date=1885&rft.aulast=Froebel&rft.aufirst=Friedrich&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feducationofman00froe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwaySloane1999" class="citation cs2">Conway, J. H.; Sloane, N. J. A. (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=upYwZ6cQumoC&pg=PA10"><i>Sphere packings, lattices and groups</i></a>, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290 (3rd ed.), New York: Springer-Verlag, p. 10, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98585-5" title="Special:BookSources/978-0-387-98585-5"><bdi>978-0-387-98585-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1662447">1662447</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sphere+packings%2C+lattices+and+groups&rft.place=New+York&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+%5BFundamental+Principles+of+Mathematical+Sciences%5D&rft.pages=10&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1999&rft.isbn=978-0-387-98585-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1662447%23id-name%3DMR&rft.aulast=Conway&rft.aufirst=J.+H.&rft.au=Sloane%2C+N.+J.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DupYwZ6cQumoC%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFPandolfini1995" class="citation cs2"><a href="/wiki/Bruce_Pandolfini" title="Bruce Pandolfini">Pandolfini, Bruce</a> (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S2gI_mExCOoC&pg=PA273"><i>Chess Thinking: The Visual Dictionary of Chess Moves, Rules, Strategies and Concepts</i></a>, Simon and Schuster, pp. 273–274, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780671795023" title="Special:BookSources/9780671795023"><bdi>9780671795023</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chess+Thinking%3A+The+Visual+Dictionary+of+Chess+Moves%2C+Rules%2C+Strategies+and+Concepts&rft.pages=273-274&rft.pub=Simon+and+Schuster&rft.date=1995&rft.isbn=9780671795023&rft.aulast=Pandolfini&rft.aufirst=Bruce&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DS2gI_mExCOoC%26pg%3DPA273&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelsohn2004" class="citation cs2">Mendelsohn, N. S. (2004), "Tiling with dominoes", <i>The College Mathematics Journal</i>, <b>35</b> (2): 115–120, <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%2F4146865">10.2307/4146865</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/4146865">4146865</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+College+Mathematics+Journal&rft.atitle=Tiling+with+dominoes&rft.volume=35&rft.issue=2&rft.pages=115-120&rft.date=2004&rft_id=info%3Adoi%2F10.2307%2F4146865&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4146865%23id-name%3DJSTOR&rft.aulast=Mendelsohn&rft.aufirst=N.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFBrucknerBrucknerThomson1997" class="citation cs2">Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1WY6u0C_jEsC&pg=PA37"><i>Real Analysis</i></a>, ClassicalRealAnalysis.com, p. 37, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-458886-5" title="Special:BookSources/978-0-13-458886-5"><bdi>978-0-13-458886-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis&rft.pages=37&rft.pub=ClassicalRealAnalysis.com&rft.date=1997&rft.isbn=978-0-13-458886-5&rft.aulast=Bruckner&rft.aufirst=Andrew+M.&rft.au=Bruckner%2C+Judith+B.&rft.au=Thomson%2C+Brian+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1WY6u0C_jEsC%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2003" class="citation cs2"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LiAlZO2ntKAC&pg=PA199"><i>Elements of Number Theory</i></a>, Springer, p. 199, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387955872" title="Special:BookSources/9780387955872"><bdi>9780387955872</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Number+Theory&rft.pages=199&rft.pub=Springer&rft.date=2003&rft.isbn=9780387955872&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLiAlZO2ntKAC%26pg%3DPA199&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLialSalzmanHestwood2005" class="citation cs2">Lial, Margaret L.; Salzman, Stanley A.; Hestwood, Diana (2005), <a rel="nofollow" class="external text" href="https://archive.org/details/basiccollegemath00lial/page/128/mode/2up"><i>Basic College Mathematics</i></a> (7th ed.), Addison Wesley, p. 128, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780321257802" title="Special:BookSources/9780321257802"><bdi>9780321257802</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+College+Mathematics&rft.pages=128&rft.edition=7th&rft.pub=Addison+Wesley&rft.date=2005&rft.isbn=9780321257802&rft.aulast=Lial&rft.aufirst=Margaret+L.&rft.au=Salzman%2C+Stanley+A.&rft.au=Hestwood%2C+Diana&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbasiccollegemath00lial%2Fpage%2F128%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFDudley1992" class="citation cs2"><a href="/wiki/Underwood_Dudley" title="Underwood Dudley">Dudley, Underwood</a> (1992), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HqeoWPsIH6EC&pg=PA242">"Perfect numbers"</a>, <a href="/wiki/Mathematical_Cranks" title="Mathematical Cranks"><i>Mathematical Cranks</i></a>, MAA Spectrum, Cambridge University Press, pp. 242–244, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780883855072" title="Special:BookSources/9780883855072"><bdi>9780883855072</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Perfect+numbers&rft.btitle=Mathematical+Cranks&rft.series=MAA+Spectrum&rft.pages=242-244&rft.pub=Cambridge+University+Press&rft.date=1992&rft.isbn=9780883855072&rft.aulast=Dudley&rft.aufirst=Underwood&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHqeoWPsIH6EC%26pg%3DPA242&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFOliveira_e_SilvaHerzogPardi2013" class="citation cs2">Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2013), <a rel="nofollow" class="external text" href="https://www.ams.org/editflow/editorial/uploads/mcom/accepted/120521-Silva/120521-Silva-v2.pdf">"Empirical verification of the even Goldbach conjecture, and computation of prime gaps, up to 4·10<sup>18</sup>"</a> <span class="cs1-format">(PDF)</span>, <i>Mathematics of Computation</i>, <b>83</b> (288): 2033–2060, <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/10.1090%2Fs0025-5718-2013-02787-1">10.1090/s0025-5718-2013-02787-1</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Empirical+verification+of+the+even+Goldbach+conjecture%2C+and+computation+of+prime+gaps%2C+up+to+4%26middot%3B10%3Csup%3E18%3C%2Fsup%3E&rft.volume=83&rft.issue=288&rft.pages=2033-2060&rft.date=2013&rft_id=info%3Adoi%2F10.1090%2Fs0025-5718-2013-02787-1&rft.aulast=Oliveira+e+Silva&rft.aufirst=Tom%C3%A1s&rft.au=Herzog%2C+Siegfried&rft.au=Pardi%2C+Silvio&rft_id=https%3A%2F%2Fwww.ams.org%2Feditflow%2Feditorial%2Fuploads%2Fmcom%2Faccepted%2F120521-Silva%2F120521-Silva-v2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>. In press.</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="CITEREFCameron1999" class="citation cs2"><a href="/wiki/Peter_Cameron_(mathematician)" title="Peter Cameron (mathematician)">Cameron, Peter J.</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4bNj8K1omGAC&pg=PA26"><i>Permutation Groups</i></a>, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, pp. 26–27, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521653787" title="Special:BookSources/9780521653787"><bdi>9780521653787</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Permutation+Groups&rft.series=London+Mathematical+Society+Student+Texts&rft.pages=26-27&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=9780521653787&rft.aulast=Cameron&rft.aufirst=Peter+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4bNj8K1omGAC%26pg%3DPA26&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFJoyner2008" class="citation cs2">Joyner, David (2008), "13.1.2 Parity conditions", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iM0fco-_Ri8C&pg=PA252"><i>Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys</i></a>, JHU Press, pp. 252–253, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801897269" title="Special:BookSources/9780801897269"><bdi>9780801897269</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=13.1.2+Parity+conditions&rft.btitle=Adventures+in+Group+Theory%3A+Rubik%27s+Cube%2C+Merlin%27s+Machine%2C+and+Other+Mathematical+Toys&rft.pages=252-253&rft.pub=JHU+Press&rft.date=2008&rft.isbn=9780801897269&rft.aulast=Joyner&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiM0fco-_Ri8C%26pg%3DPA252&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFBenderGlauberman1994" class="citation cs2">Bender, Helmut; Glauberman, George (1994), <i>Local analysis for the odd order theorem</i>, London Mathematical Society Lecture Note Series, vol. 188, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-45716-3" title="Special:BookSources/978-0-521-45716-3"><bdi>978-0-521-45716-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1311244">1311244</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Local+analysis+for+the+odd+order+theorem&rft.place=Cambridge&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-45716-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1311244%23id-name%3DMR&rft.aulast=Bender&rft.aufirst=Helmut&rft.au=Glauberman%2C+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeterfalvi2000" class="citation cs2">Peterfalvi, Thomas (2000), <i>Character theory for the odd order theorem</i>, London Mathematical Society Lecture Note Series, vol. 272, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-64660-4" title="Special:BookSources/978-0-521-64660-4"><bdi>978-0-521-64660-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1747393">1747393</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Character+theory+for+the+odd+order+theorem&rft.place=Cambridge&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-64660-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1747393%23id-name%3DMR&rft.aulast=Peterfalvi&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFGustafsonHughes2012" class="citation cs2">Gustafson, Roy David; Hughes, Jeffrey D. (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sxZpddk1fTIC&pg=PA315"><i>College Algebra</i></a> (11th ed.), Cengage Learning, p. 315, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781111990909" title="Special:BookSources/9781111990909"><bdi>9781111990909</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Algebra&rft.pages=315&rft.edition=11th&rft.pub=Cengage+Learning&rft.date=2012&rft.isbn=9781111990909&rft.aulast=Gustafson&rft.aufirst=Roy+David&rft.au=Hughes%2C+Jeffrey+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsxZpddk1fTIC%26pg%3DPA315&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFJainIyengar2007" class="citation cs2">Jain, R. K.; Iyengar, S. R. K. (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=crOxJNLE5psC&pg=PA853"><i>Advanced Engineering Mathematics</i></a>, Alpha Science Int'l Ltd., p. 853, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781842651858" title="Special:BookSources/9781842651858"><bdi>9781842651858</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Engineering+Mathematics&rft.pages=853&rft.pub=Alpha+Science+Int%27l+Ltd.&rft.date=2007&rft.isbn=9781842651858&rft.aulast=Jain&rft.aufirst=R.+K.&rft.au=Iyengar%2C+S.+R.+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcrOxJNLE5psC%26pg%3DPA853&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFGuy1996" class="citation cs2"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (1996), "Impartial games", <i>Games of no chance (Berkeley, CA, 1994)</i>, Math. Sci. Res. Inst. Publ., vol. 29, Cambridge: Cambridge Univ. Press, pp. 61–78, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1427957">1427957</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Impartial+games&rft.btitle=Games+of+no+chance+%28Berkeley%2C+CA%2C+1994%29&rft.place=Cambridge&rft.series=Math.+Sci.+Res.+Inst.+Publ.&rft.pages=61-78&rft.pub=Cambridge+Univ.+Press&rft.date=1996&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1427957%23id-name%3DMR&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cYB-ra2T8i4C&pg=PA68">p. 68</a>.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernhardt2009" class="citation cs2">Bernhardt, Chris (2009), <a rel="nofollow" class="external text" href="https://digitalcommons.fairfield.edu/content_policy.pdf">"Evil twins alternate with odious twins"</a> <span class="cs1-format">(PDF)</span>, <i>Mathematics Magazine</i>, <b>82</b> (1): 57–62, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F193009809x469084">10.4169/193009809x469084</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/27643161">27643161</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Evil+twins+alternate+with+odious+twins&rft.volume=82&rft.issue=1&rft.pages=57-62&rft.date=2009&rft_id=info%3Adoi%2F10.4169%2F193009809x469084&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27643161%23id-name%3DJSTOR&rft.aulast=Bernhardt&rft.aufirst=Chris&rft_id=https%3A%2F%2Fdigitalcommons.fairfield.edu%2Fcontent_policy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoserChen2012" class="citation cs2">Moser, Stefan M.; Chen, Po-Ning (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gFhJXsGXNj8C&pg=PA19"><i>A Student's Guide to Coding and Information Theory</i></a>, Cambridge University Press, pp. 19–20, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107015838" title="Special:BookSources/9781107015838"><bdi>9781107015838</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+Student%27s+Guide+to+Coding+and+Information+Theory&rft.pages=19-20&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=9781107015838&rft.aulast=Moser&rft.aufirst=Stefan+M.&rft.au=Chen%2C+Po-Ning&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgFhJXsGXNj8C%26pg%3DPA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFBerrou2011" class="citation cs2">Berrou, Claude (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZLPWNq8JN9QC&pg=PA4"><i>Codes and turbo codes</i></a>, Springer, p. 4, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9782817800394" title="Special:BookSources/9782817800394"><bdi>9782817800394</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Codes+and+turbo+codes&rft.pages=4&rft.pub=Springer&rft.date=2011&rft.isbn=9782817800394&rft.aulast=Berrou&rft.aufirst=Claude&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZLPWNq8JN9QC%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFRandall2005" class="citation cs2">Randall, Robert H. (2005), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l9pO7vAvLpUC&pg=PA181"><i>An Introduction to Acoustics</i></a>, Dover, p. 181, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486442518" title="Special:BookSources/9780486442518"><bdi>9780486442518</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+Acoustics&rft.pages=181&rft.pub=Dover&rft.date=2005&rft.isbn=9780486442518&rft.aulast=Randall&rft.aufirst=Robert+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dl9pO7vAvLpUC%26pg%3DPA181&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFCromleyMcLafferty2011" class="citation cs2">Cromley, Ellen K.; McLafferty, Sara L. (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LeaEPg9vCrsC&pg=PA100"><i>GIS and Public Health</i></a> (2nd ed.), Guilford Press, p. 100, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781462500628" title="Special:BookSources/9781462500628"><bdi>9781462500628</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=GIS+and+Public+Health&rft.pages=100&rft.edition=2nd&rft.pub=Guilford+Press&rft.date=2011&rft.isbn=9781462500628&rft.aulast=Cromley&rft.aufirst=Ellen+K.&rft.au=McLafferty%2C+Sara+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLeaEPg9vCrsC%26pg%3DPA100&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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="CITEREFSwift2011" class="citation cs2">Swift, Earl (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=59dQ_rwoh3UC&pg=PA95"><i>The Big Roads: The Untold Story of the Engineers, Visionaries, and Trailblazers Who Created the American Superhighways</i></a>, Houghton Mifflin Harcourt, p. 95, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780547549132" title="Special:BookSources/9780547549132"><bdi>9780547549132</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+Big+Roads%3A+The+Untold+Story+of+the+Engineers%2C+Visionaries%2C+and+Trailblazers+Who+Created+the+American+Superhighways&rft.pages=95&rft.pub=Houghton+Mifflin+Harcourt&rft.date=2011&rft.isbn=9780547549132&rft.aulast=Swift&rft.aufirst=Earl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D59dQ_rwoh3UC%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" 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 id="CITEREFLauer2010" class="citation cs2">Lauer, Chris (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NpZbEihL0ZgC&pg=PA90"><i>Southwest Airlines</i></a>, Corporations that changed the world, ABC-CLIO, p. 90, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780313378638" title="Special:BookSources/9780313378638"><bdi>9780313378638</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Southwest+Airlines&rft.series=Corporations+that+changed+the+world&rft.pages=90&rft.pub=ABC-CLIO&rft.date=2010&rft.isbn=9780313378638&rft.aulast=Lauer&rft.aufirst=Chris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNpZbEihL0ZgC%26pg%3DPA90&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParity+%28mathematics%29" class="Z3988"></span>.</span> </li> </ol></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=Parity_(mathematics)&oldid=1259783707">https://en.wikipedia.org/w/index.php?title=Parity_(mathematics)&oldid=1259783707</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Parity_(mathematics)" title="Category:Parity (mathematics)">Parity (mathematics)</a></li><li><a href="/wiki/Category:Elementary_arithmetic" title="Category:Elementary arithmetic">Elementary arithmetic</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></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 27 November 2024, at 01:01<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=Parity_(mathematics)&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-5cd4cd96d5-4vn7k","wgBackendResponseTime":229,"wgPageParseReport":{"limitreport":{"cputime":"0.430","walltime":"0.624","ppvisitednodes":{"value":2102,"limit":1000000},"postexpandincludesize":{"value":69718,"limit":2097152},"templateargumentsize":{"value":1415,"limit":2097152},"expansiondepth":{"value":8,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":109938,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 419.063 1 -total"," 60.79% 254.753 1 Template:Reflist"," 53.01% 222.126 30 Template:Citation"," 17.03% 71.376 1 Template:Short_description"," 10.14% 42.491 2 Template:Pagetype"," 9.43% 39.503 1 Template:Other_uses"," 3.97% 16.625 5 Template:Main_other"," 3.25% 13.617 1 Template:SDcat"," 2.75% 11.523 1 Template:Chess_diagram"," 2.32% 9.726 1 Template:Blockquote"]},"scribunto":{"limitreport-timeusage":{"value":"0.253","limit":"10.000"},"limitreport-memusage":{"value":5773168,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-5f67bcf949-4v6dn","timestamp":"20241127010130","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Parity (mathematics)","url":"https:\/\/en.wikipedia.org\/wiki\/Parity_(mathematics)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q230967","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q230967","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-11-02T00:08:55Z","dateModified":"2024-11-27T01:01:25Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e1\/Parity_of_5_and_6_Cuisenaire_rods.png","headline":"property of being an odd or even number"}</script> </body> </html>