CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Complex number - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"3830f6d2-fd8e-4b4c-9912-012812bbe8bc","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Complex_number","wgTitle":"Complex number","wgCurRevisionId":1281498490,"wgRevisionId":1281498490,"wgArticleId":5826,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 Latin-language sources (la)","CS1 Danish-language sources (da)","CS1 French-language sources (fr)","CS1: unfit URL","CS1 German-language sources (de)","Articles with short description","Short description is different from Wikidata","Wikipedia indefinitely move-protected pages","Use dmy dates from June 2020","Commons category link is on Wikidata","Pages that use a deprecated format of the math tags","Composition algebras","Complex numbers"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Complex_number","wgRelevantArticleId":5826,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":["sysop"],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":90000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q11567","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.21"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/1200px-A_plus_bi.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="804"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/800px-A_plus_bi.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="536"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/640px-A_plus_bi.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="429"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Complex number - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Complex_number"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Complex_number&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/Complex_number"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Complex_number rootpage-Complex_number skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </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">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</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">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</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">Recent changes</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">Upload file</a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages">Special pages</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"> <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;"> </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"> Search </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" > </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" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class="">Donate</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=Complex+number" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</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=Complex+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</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" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Complex+number" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Complex+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</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">learn more</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">Contributions</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">Talk</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_and_basic_operations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_and_basic_operations"> <div class="vector-toc-text"> 1 Definition and basic operations </div> </a> <button aria-controls="toc-Definition_and_basic_operations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition and basic operations subsection </button> <ul id="toc-Definition_and_basic_operations-sublist" class="vector-toc-list"> <li id="toc-Addition_and_subtraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition_and_subtraction"> <div class="vector-toc-text"> 1.1 Addition and subtraction </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"> 1.2 Multiplication </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_conjugate,_absolute_value,_argument_and_division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_conjugate,_absolute_value,_argument_and_division"> <div class="vector-toc-text"> 1.3 Complex conjugate, absolute value, argument and division </div> </a> <ul id="toc-Complex_conjugate,_absolute_value,_argument_and_division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polar_form"> <div class="vector-toc-text"> 1.4 Polar form </div> </a> <ul id="toc-Polar_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Powers_and_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Powers_and_roots"> <div class="vector-toc-text"> 1.5 Powers and roots </div> </a> <ul id="toc-Powers_and_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fundamental_theorem_of_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_theorem_of_algebra"> <div class="vector-toc-text"> 1.6 Fundamental theorem of algebra </div> </a> <ul id="toc-Fundamental_theorem_of_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_algebraic_aspects" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Abstract_algebraic_aspects"> <div class="vector-toc-text"> 3 Abstract algebraic aspects </div> </a> <button aria-controls="toc-Abstract_algebraic_aspects-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Abstract algebraic aspects subsection </button> <ul id="toc-Abstract_algebraic_aspects-sublist" class="vector-toc-list"> <li id="toc-Construction_as_a_quotient_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_as_a_quotient_field"> <div class="vector-toc-text"> 3.1 Construction as a quotient field </div> </a> <ul id="toc-Construction_as_a_quotient_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_representation_of_complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_representation_of_complex_numbers"> <div class="vector-toc-text"> 3.2 Matrix representation of complex numbers </div> </a> <ul id="toc-Matrix_representation_of_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_analysis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_analysis"> <div class="vector-toc-text"> 4 Complex analysis </div> </a> <button aria-controls="toc-Complex_analysis-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Complex analysis subsection </button> <ul id="toc-Complex_analysis-sublist" class="vector-toc-list"> <li id="toc-Convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergence"> <div class="vector-toc-text"> 4.1 Convergence </div> </a> <ul id="toc-Convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_exponential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_exponential"> <div class="vector-toc-text"> 4.2 Complex exponential </div> </a> <ul id="toc-Complex_exponential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_logarithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_logarithm"> <div class="vector-toc-text"> 4.3 Complex logarithm </div> </a> <ul id="toc-Complex_logarithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_sine_and_cosine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_sine_and_cosine"> <div class="vector-toc-text"> 4.4 Complex sine and cosine </div> </a> <ul id="toc-Complex_sine_and_cosine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Holomorphic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Holomorphic_functions"> <div class="vector-toc-text"> 4.5 Holomorphic functions </div> </a> <ul id="toc-Holomorphic_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> 5.1 Geometry </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> <li id="toc-Shapes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shapes"> <div class="vector-toc-text"> 5.1.1 Shapes </div> </a> <ul id="toc-Shapes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractal_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fractal_geometry"> <div class="vector-toc-text"> 5.1.2 Fractal geometry </div> </a> <ul id="toc-Fractal_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Triangles"> <div class="vector-toc-text"> 5.1.3 Triangles </div> </a> <ul id="toc-Triangles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebraic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_number_theory"> <div class="vector-toc-text"> 5.2 Algebraic number theory </div> </a> <ul id="toc-Algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytic_number_theory"> <div class="vector-toc-text"> 5.3 Analytic number theory </div> </a> <ul id="toc-Analytic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Improper_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Improper_integrals"> <div class="vector-toc-text"> 5.4 Improper integrals </div> </a> <ul id="toc-Improper_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamic_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamic_equations"> <div class="vector-toc-text"> 5.5 Dynamic equations </div> </a> <ul id="toc-Dynamic_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_algebra"> <div class="vector-toc-text"> 5.6 Linear algebra </div> </a> <ul id="toc-Linear_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_applied_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_applied_mathematics"> <div class="vector-toc-text"> 5.7 In applied mathematics </div> </a> <ul id="toc-In_applied_mathematics-sublist" class="vector-toc-list"> <li id="toc-Control_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Control_theory"> <div class="vector-toc-text"> 5.7.1 Control theory </div> </a> <ul id="toc-Control_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signal_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Signal_analysis"> <div class="vector-toc-text"> 5.7.2 Signal analysis </div> </a> <ul id="toc-Signal_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_physics"> <div class="vector-toc-text"> 5.8 In physics </div> </a> <ul id="toc-In_physics-sublist" class="vector-toc-list"> <li id="toc-Electromagnetism_and_electrical_engineering" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electromagnetism_and_electrical_engineering"> <div class="vector-toc-text"> 5.8.1 Electromagnetism and electrical engineering </div> </a> <ul id="toc-Electromagnetism_and_electrical_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_dynamics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fluid_dynamics"> <div class="vector-toc-text"> 5.8.2 Fluid dynamics </div> </a> <ul id="toc-Fluid_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 5.8.3 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relativity"> <div class="vector-toc-text"> 5.8.4 Relativity </div> </a> <ul id="toc-Relativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Characterizations,_generalizations_and_related_notions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Characterizations,_generalizations_and_related_notions"> <div class="vector-toc-text"> 6 Characterizations, generalizations and related notions </div> </a> <button aria-controls="toc-Characterizations,_generalizations_and_related_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Characterizations, generalizations and related notions subsection </button> <ul id="toc-Characterizations,_generalizations_and_related_notions-sublist" class="vector-toc-list"> <li id="toc-Algebraic_characterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_characterization"> <div class="vector-toc-text"> 6.1 Algebraic characterization </div> </a> <ul id="toc-Algebraic_characterization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization_as_a_topological_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_as_a_topological_field"> <div class="vector-toc-text"> 6.2 Characterization as a topological field </div> </a> <ul id="toc-Characterization_as_a_topological_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_number_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_number_systems"> <div class="vector-toc-text"> 6.3 Other number systems </div> </a> <ul id="toc-Other_number_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> 9.1 Historical references </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Complex number</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 132 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-132" aria-hidden="true" > 132 languages </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/Komplekse_getal" title="Komplekse getal – Afrikaans" lang="af" hreflang="af" data-title="Komplekse getal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Komplexe Zahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%A0%E1%89%85%E1%8C%A3%E1%8C%AB_%E1%89%81%E1%8C%A5%E1%88%AD" title="የአቅጣጫ ቁጥር – Amharic" lang="am" hreflang="am" data-title="የአቅጣጫ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kompleksloho" title="Kompleksloho – Inari Sami" lang="smn" hreflang="smn" data-title="Kompleksloho" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target">Anarâškielâ</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – Angika" lang="anp" hreflang="anp" data-title="समिश्र संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%B1%D9%83%D8%A8" title="عدد مركب – Arabic" lang="ar" hreflang="ar" data-title="عدد مركب" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_complexo" title="Numero complexo – Aragonese" lang="an" hreflang="an" data-title="Numero complexo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – Assamese" lang="as" hreflang="as" data-title="জটিল সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_complexu" title="Númberu complexu – Asturian" lang="ast" hreflang="ast" data-title="Númberu complexu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapy_rypy%27%C5%A9" title="Papapy rypy'ũ – Guarani" lang="gn" hreflang="gn" data-title="Papapy rypy'ũ" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kompleks_%C9%99d%C9%99dl%C9%99r" title="Kompleks ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Kompleks ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D9%85%D9%BE%D9%84%DA%A9%D8%B3_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="کومپلکس ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="کومپلکس ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="জটিল সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ho%CC%8Dk-cha%CC%8Dp-s%C3%B2%CD%98" title="Ho̍k-cha̍p-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Ho̍k-cha̍p-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D2%BB%D0%B0%D0%BD" title="Комплекслы һан – Bashkir" lang="ba" hreflang="ba" data-title="Комплекслы һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – Belarusian" lang="be" hreflang="be" data-title="Камплексны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Камплексны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексно число – Bulgarian" lang="bg" hreflang="bg" data-title="Комплексно число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – Bosnian" lang="bs" hreflang="bs" data-title="Kompleksan broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Комплекс тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_complex" title="Nombre complex – Catalan" lang="ca" hreflang="ca" data-title="Nombre complex" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Комплекслă хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Комплекслă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Komplexn%C3%AD_%C4%8D%C3%ADslo" title="Komplexní číslo – Czech" lang="cs" hreflang="cs" data-title="Komplexní číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cymhlyg" title="Rhif cymhlyg – Welsh" lang="cy" hreflang="cy" data-title="Rhif cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://da.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – Danish" lang="da" hreflang="da" data-title="Komplekse tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – German" lang="de" hreflang="de" data-title="Komplexe Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – Estonian" lang="et" hreflang="et" data-title="Kompleksarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Μιγαδικός αριθμός – Greek" lang="el" hreflang="el" data-title="Μιγαδικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_cumpl%C3%AAs" title="Nómmer cumplês – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer cumplês" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_complejo" title="Número complejo – Spanish" lang="es" hreflang="es" data-title="Número complejo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_nombro" title="Kompleksa nombro – Esperanto" lang="eo" hreflang="eo" data-title="Kompleksa nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_konplexu" title="Zenbaki konplexu – Basque" lang="eu" hreflang="eu" data-title="Zenbaki konplexu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%AE%D8%AA%D9%84%D8%B7" title="عدد مختلط – Persian" lang="fa" hreflang="fa" data-title="عدد مختلط" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Jatil_ginti" title="Jatil ginti – Fiji Hindi" lang="hif" hreflang="hif" data-title="Jatil ginti" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Komplekst_tal" title="Komplekst tal – Faroese" lang="fo" hreflang="fo" data-title="Komplekst tal" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_complexe" title="Nombre complexe – French" lang="fr" hreflang="fr" data-title="Nombre complexe" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kompleks_getal" title="Kompleks getal – Western Frisian" lang="fy" hreflang="fy" data-title="Kompleks getal" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_choimpl%C3%A9ascach" title="Uimhir choimpléascach – Irish" lang="ga" hreflang="ga" data-title="Uimhir choimpléascach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – Galician" lang="gl" hreflang="gl" data-title="Número complexo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Gan" lang="gan" hreflang="gan" data-title="複數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%95%E0%AA%B0_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE%E0%AA%93" title="સંકર સંખ્યાઓ – Gujarati" lang="gu" hreflang="gu" data-title="સંકર સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B8%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Комплексин тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Комплексин тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수 – Korean" lang="ko" hreflang="ko" data-title="복소수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A9%D5%AB%D5%BE" title="Կոմպլեքս թիվ – Armenian" lang="hy" hreflang="hy" data-title="Կոմպլեքս թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</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%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – Hindi" lang="hi" hreflang="hi" data-title="समिश्र संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kompleksni_broj" title="Kompleksni broj – Croatian" lang="hr" hreflang="hr" data-title="Kompleksni broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Komplexa_nombro" title="Komplexa nombro – Ido" lang="io" hreflang="io" data-title="Komplexa nombro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_kompleks" title="Bilangan kompleks – Indonesian" lang="id" hreflang="id" data-title="Bilangan kompleks" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_complexe" title="Numero complexe – Interlingua" lang="ia" hreflang="ia" data-title="Numero complexe" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BE%D0%BD_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Комплексон нымæц – Ossetic" lang="os" hreflang="os" data-title="Комплексон нымæц" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tvinnt%C3%B6lur" title="Tvinntölur – Icelandic" lang="is" hreflang="is" data-title="Tvinntölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_complesso" title="Numero complesso – Italian" lang="it" hreflang="it" data-title="Numero complesso" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A8%D7%95%D7%9B%D7%91" title="מספר מרוכב – Hebrew" lang="he" hreflang="he" data-title="מספר מרוכב" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Nd%C9%A9_nd%C9%A9_%C3%B1%CA%8A%C5%8B" title="Ndɩ ndɩ ñʊŋ – Kabiye" lang="kbp" hreflang="kbp" data-title="Ndɩ ndɩ ñʊŋ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target">Kabɩyɛ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A1%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="კომპლექსური რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="კომპლექსური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%88%D0%B5%D0%BD_%D1%81%D0%B0%D0%BD" title="Кешен сан – Kazakh" lang="kk" hreflang="kk" data-title="Кешен сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niver_kompleth" title="Niver kompleth – Cornish" lang="kw" hreflang="kw" data-title="Niver kompleth" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_changamano" title="Namba changamano – Swahili" lang="sw" hreflang="sw" data-title="Namba changamano" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_kompleks" title="Nonm kompleks – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm kompleks" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D1%82%D2%AF%D2%AF_%D1%81%D0%B0%D0%BD" title="Комплекстүү сан – Kyrgyz" lang="ky" hreflang="ky" data-title="Комплекстүү сан" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AA%E0%BA%BB%E0%BA%99" title="ຈຳນວນສົນ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນສົນ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_complexus" title="Numerus complexus – Latin" lang="la" hreflang="la" data-title="Numerus complexus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komplekss_skaitlis" title="Komplekss skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Komplekss skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kompleksinis_skai%C4%8Dius" title="Kompleksinis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Kompleksinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Complex_getal" title="Complex getal – Limburgish" lang="li" hreflang="li" data-title="Complex getal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/relcimdyna%27u" title="relcimdyna'u – Lojban" lang="jbo" hreflang="jbo" data-title="relcimdyna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_compless" title="Numer compless – Lombard" lang="lmo" hreflang="lmo" data-title="Numer compless" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Komplex_sz%C3%A1mok" title="Komplex számok – Hungarian" lang="hu" hreflang="hu" data-title="Komplex számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексен број – Macedonian" lang="mk" hreflang="mk" data-title="Комплексен број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_haro" title="Isa haro – Malagasy" lang="mg" hreflang="mg" data-title="Isa haro" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B0%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="മിശ്രസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="മിശ്രസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संमिश्र संख्या – Marathi" lang="mr" hreflang="mr" data-title="संमिश्र संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_kompleks" title="Nombor kompleks – Malay" lang="ms" hreflang="ms" data-title="Nombor kompleks" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – Mongolian" lang="mn" hreflang="mn" data-title="Комплекс тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%94%E1%80%BA%E1%80%95%E1%80%9C%E1%80%80%E1%80%BA%E1%80%85%E1%80%BA%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="ကွန်ပလက်စ်ကိန်း – Burmese" lang="my" hreflang="my" data-title="ကွန်ပလက်စ်ကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Complex_getal" title="Complex getal – Dutch" lang="nl" hreflang="nl" data-title="Complex getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E6%95%B0" title="複素数 – Japanese" lang="ja" hreflang="ja" data-title="複素数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kompleks_taal" title="Kompleks taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Kompleks taal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Komplekst_tall" title="Komplekst tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Komplekst tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Komplekse tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_compl%C3%A8xe" title="Nombre complèxe – Occitan" lang="oc" hreflang="oc" data-title="Nombre complèxe" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Xaxxamaa" title="Lakkoofsa Xaxxamaa – Oromo" lang="om" hreflang="om" data-title="Lakkoofsa Xaxxamaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kompleks_sonlar" title="Kompleks sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Kompleks sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AA%E0%A8%B2%E0%A9%88%E0%A8%95%E0%A8%B8_%E0%A8%A8%E0%A9%B0%E0%A8%AC%E0%A8%B0" title="ਕੰਪਲੈਕਸ ਨੰਬਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੰਪਲੈਕਸ ਨੰਬਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%85%D9%BE%D9%84%DB%8C%DA%A9%D8%B3_%D9%86%D9%85%D8%A8%D8%B1" title="کمپلیکس نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کمپلیکس نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Komplex_nomba" title="Komplex nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Komplex nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%80%E1%9E%BB%E1%9F%86%E1%9E%95%E1%9F%92%E1%9E%9B%E1%9E%B7%E1%9E%85" title="ចំនួនកុំផ្លិច – Khmer" lang="km" hreflang="km" data-title="ចំនួនកុំផ្លិច" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_compless" title="Nùmer compless – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer compless" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Komplexe_Tall" title="Komplexe Tall – Low German" lang="nds" hreflang="nds" data-title="Komplexe Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_zespolone" title="Liczby zespolone – Polish" lang="pl" hreflang="pl" data-title="Liczby zespolone" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – Portuguese" lang="pt" hreflang="pt" data-title="Número complexo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_complex" title="Număr complex – Romanian" lang="ro" hreflang="ro" data-title="Număr complex" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D1%96%D1%81%D0%BB%D0%BE" title="Комплексне чісло – Rusyn" lang="rue" hreflang="rue" data-title="Комплексне чісло" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексное число – Russian" lang="ru" hreflang="ru" data-title="Комплексное число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D0%B0%D1%85%D1%81%D0%B0%D0%B0%D0%BD" title="Комплекс ахсаан – Yakut" lang="sah" hreflang="sah" data-title="Комплекс ахсаан" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Complex_nummer" title="Complex nummer – Scots" lang="sco" hreflang="sco" data-title="Complex nummer" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri_kompleks" title="Numri kompleks – Albanian" lang="sq" hreflang="sq" data-title="Numri kompleks" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_cumplessu" title="Nùmmuru cumplessu – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru cumplessu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9A%E0%B7%93%E0%B6%BB%E0%B7%8A%E0%B6%AB_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="සංකීර්ණ සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="සංකීර්ණ සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Complex_number" title="Complex number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Complex number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Komplexn%C3%A9_%C4%8D%C3%ADslo" title="Komplexné číslo – Slovak" lang="sk" hreflang="sk" data-title="Komplexné číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kompleksno_%C5%A1tevilo" title="Kompleksno število – Slovenian" lang="sl" hreflang="sl" data-title="Kompleksno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_kakan" title="Thiin kakan – Somali" lang="so" hreflang="so" data-title="Thiin kakan" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%A6%D8%A7%D9%88%DB%8E%D8%AA%DB%95" title="ژمارەی ئاوێتە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی ئاوێتە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексан број – Serbian" lang="sr" hreflang="sr" data-title="Комплексан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kompleksan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kompleksiluku" title="Kompleksiluku – Finnish" lang="fi" hreflang="fi" data-title="Kompleksiluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komplexa_tal" title="Komplexa tal – Swedish" lang="sv" hreflang="sv" data-title="Komplexa tal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Komplikadong_bilang" title="Komplikadong bilang – Tagalog" lang="tl" hreflang="tl" data-title="Komplikadong bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%86%E0%AE%A3%E0%AF%8D" title="சிக்கலெண் – Tamil" lang="ta" hreflang="ta" data-title="சிக்கலெண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Am%E1%B8%8Dan_asemlal" title="Amḍan asemlal – Kabyle" lang="kab" hreflang="kab" data-title="Amḍan asemlal" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%81%D0%B0%D0%BD" title="Комплекс сан – Tatar" lang="tt" hreflang="tt" data-title="Комплекс сан" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%82%E0%B0%95%E0%B1%80%E0%B0%B0%E0%B1%8D%E0%B0%A3_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="సంకీర్ణ సంఖ్యలు – Telugu" lang="te" hreflang="te" data-title="సంకీర్ణ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99" title="จำนวนเชิงซ้อน – Thai" lang="th" hreflang="th" data-title="จำนวนเชิงซ้อน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</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%D0%B8_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D3%A3" title="Адади комплексӣ – Tajik" lang="tg" hreflang="tg" data-title="Адади комплексӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karma%C5%9F%C4%B1k_say%C4%B1" title="Karmaşık sayı – Turkish" lang="tr" hreflang="tr" data-title="Karmaşık sayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексне число – Ukrainian" lang="uk" hreflang="uk" data-title="Комплексне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AE%D9%84%D9%88%D8%B7_%D8%B9%D8%AF%D8%AF" title="مخلوط عدد – Urdu" lang="ur" hreflang="ur" data-title="مخلوط عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Numaro_conpleso" title="Numaro conpleso – Venetian" lang="vec" hreflang="vec" data-title="Numaro conpleso" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ph%E1%BB%A9c" title="Số phức – Vietnamese" lang="vi" hreflang="vi" data-title="Số phức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – Võro" lang="vro" hreflang="vro" data-title="Kompleksarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="複數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Complexe_getalln" title="Complexe getalln – West Flemish" lang="vls" hreflang="vls" data-title="Complexe getalln" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Komplikado_nga_ihap" title="Komplikado nga ihap – Waray" lang="war" hreflang="war" data-title="Komplikado nga ihap" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="复数（数学） – Wu" lang="wuu" hreflang="wuu" data-title="复数（数学）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%9E%D7%A4%D7%9C%D7%A2%D7%A7%D7%A1%D7%A2_%D7%A6%D7%90%D7%9C" title="קאמפלעקסע צאל – Yiddish" lang="yi" hreflang="yi" data-title="קאמפלעקסע צאל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</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_t%C3%B3%E1%B9%A3%C3%B2ro" title="Nọ́mbà tóṣòro – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà tóṣòro" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Cantonese" lang="yue" hreflang="yue" data-title="複數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_kompleks" title="Amaro kompleks – Dimli" lang="diq" hreflang="diq" data-title="Amaro kompleks" data-language-autonym="Zazaki" data-language-local-name="Dimli" class="interlanguage-link-target">Zazaki</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Kuompleks%C4%97nis_skaitlios" title="Kuompleksėnis skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Kuompleksėnis skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学) – Chinese" lang="zh" hreflang="zh" data-title="复数 (数学)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_kompleks" title="Lumur kompleks – Iban" lang="iba" hreflang="iba" data-title="Lumur kompleks" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target">Jaku Iban</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11567#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></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/Complex_number" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Complex_number" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</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" >English </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/Complex_number">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Complex_number&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Complex_number&action=history" title="Past revisions of this page [h]" accesskey="h">View history</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" >Tools </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/Complex_number">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Complex_number&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Complex_number&action=history">View history</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/Complex_number" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Complex_number" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Complex_number&oldid=1281498490" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Complex_number&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Complex_number&id=1281498490&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</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%2FComplex_number">Get shortened URL</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%2FComplex_number">Download QR code</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=Complex_number&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Complex_number&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</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:Complex_numbers" hreflang="en">Wikimedia Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Algebra/Chapter_20" hreflang="en">Wikibooks</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/Q11567" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</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"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Number with a real and an imaginary part</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:A_plus_bi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/250px-A_plus_bi.svg.png" decoding="async" width="250" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/375px-A_plus_bi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/A_plus_bi.svg/500px-A_plus_bi.svg.png 2x" data-file-width="182" data-file-height="122" /></a><figcaption>A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an <a href="/wiki/Argand_diagram" class="mw-redirect" title="Argand diagram">Argand diagram</a>, representing the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. Re is the real axis, Im is the imaginary axis, and i is the "<a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>", that satisfies i2 = −1.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a complex number is an element of a <a href="/wiki/Number_system" class="mw-redirect" title="Number system">number system</a> that extends the <a href="/wiki/Real_number" title="Real number">real numbers</a> with a specific element denoted i, called the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> and satisfying the <a href="/wiki/Equation" title="Equation">equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}" />; every complex number can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}" />, where a and b are real numbers. Because no real number satisfies the above equation, i was called an <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary number</a> by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>. For the complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}" />, a is called the <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>real part, and b is called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509" />imaginary part. The set of complex numbers is denoted by either of the symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> or C. Despite the historical nomenclature, "imaginary" complex numbers have a <a href="/wiki/Mathematical_sciences" title="Mathematical sciences">mathematical</a> existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> Complex numbers allow solutions to all <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a>, even those that have no solutions in real numbers. More precisely, the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+1)^{2}=-9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)^{2}=-9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c58fbc3ad9fa5d3c0e77886c766272117716ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.265ex; height:3.176ex;" alt="{\displaystyle (x+1)^{2}=-9}" /> has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1+3i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1+3i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d89e7924fb569a417b4bac3c2e433d9d43c06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.776ex; height:2.343ex;" alt="{\displaystyle -1+3i}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1-3i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1-3i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58cd7219c9765a1f1630789ef699aca94c01de79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.776ex; height:2.343ex;" alt="{\displaystyle -1-3i}" />. Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}" /> along with the <a href="/wiki/Associative_law" class="mw-redirect" title="Associative law">associative</a>, <a href="/wiki/Commutative_law" class="mw-redirect" title="Commutative law">commutative</a>, and <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive laws</a>. Every nonzero complex number has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>. This makes the complex numbers a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> with the real numbers as a subfield. Because of these properties, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c719d89004896f1109a1cffd87d8eef7ee6a89cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.839ex; height:2.343ex;" alt="{\displaystyle a+bi=a+ib}" />⁠, and which form is written depends upon convention and style considerations. The complex numbers also form a <a href="/wiki/Real_vector_space" class="mw-redirect" title="Real vector space">real vector space</a> of <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">dimension two</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,i\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,i\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d207003fe316ebadc00de228a4e93ea13bec2fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.324ex; height:2.843ex;" alt="{\displaystyle \{1,i\}}" /> as a <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a>. This standard basis makes the complex numbers a <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>, called the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, which is pictured as the horizontal axis of the complex plane, while real multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /> are the vertical axis. A complex number can also be defined by its geometric <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinates</a>: the radius is called the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>. Adding a fixed complex number to all complex numbers defines a <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> in the complex plane, and multiplying by a fixed complex number is a <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a> centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a> is the <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> with respect to the real axis. The complex numbers form a rich structure that is simultaneously an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>, a <a href="/wiki/Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">commutative algebra</a> over the reals, and a <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a> of dimension two. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Definition_and_basic_operations">Definition and basic operations</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=1" title="Edit section: Definition and basic operations">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_numbers_intheplane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Complex_numbers_intheplane.svg/220px-Complex_numbers_intheplane.svg.png" decoding="async" width="220" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Complex_numbers_intheplane.svg/330px-Complex_numbers_intheplane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Complex_numbers_intheplane.svg/440px-Complex_numbers_intheplane.svg.png 2x" data-file-width="218" data-file-height="215" /></a><figcaption>Various complex numbers depicted in the complex plane.</figcaption></figure> A complex number is an expression <a href="/wiki/Of_the_form" title="Of the form">of the form</a> a + bi, where a and b are <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, and i is an abstract symbol, the so-called <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, whose meaning will be explained further below. For example, 2 + 3i is a complex number.<a href="#cite_note-3">[3]</a> For a complex number a + bi, the real number a is called its real part, and the real number b (not the complex number bi) is its imaginary part.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> The real part of a complex number z is denoted Re(z), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Re}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">e</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Re}}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6f619d9e198cf009e4c24ae6e85542eab8c1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.952ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Re}}(z)}" />, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f74374916b6b384e5b62e3ede892b09254db277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {R}}(z)}" />; the imaginary part is Im(z), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Im}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Im}}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208a9a6f3d4af3ec5ecb7f2c22d69d4815128b76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.069ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Im}}(z)}" />, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {I}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">I</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {I}}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ce588de4a48ca371d71504ae9afe6906022378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.185ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {I}}(z)}" />: for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Re} (2+3i)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Re} (2+3i)=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76c142d59b1eefb02daf000d9e5d963452636ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.781ex; height:2.843ex;" alt="{\textstyle \operatorname {Re} (2+3i)=2}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (2+3i)=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (2+3i)=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88150bbeaa659f1e1085e19a733f9925063f597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.813ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (2+3i)=3}" />. A complex number z can be identified with the <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Re (z),\Im (z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">ℑ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Re (z),\Im (z))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf496ff70298f00e695abe890c7ede154eb0a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.85ex; height:2.843ex;" alt="{\displaystyle (\Re (z),\Im (z))}" />, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> or <a href="/wiki/Argand_diagram" class="mw-redirect" title="Argand diagram">Argand diagram</a>.<a href="#cite_note-6">[6]</a><a href="#cite_note-:2-7">[7]</a><a href="#cite_note-8">[a]</a> The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_number_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/220px-Complex_number_illustration.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/330px-Complex_number_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/440px-Complex_number_illustration.svg.png 2x" data-file-width="180" data-file-height="180" /></a><figcaption>A complex number z, as a point (black) and its <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">position vector</a> (blue).</figcaption></figure> A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary number</a> bi is a complex number 0 + bi, whose real part is zero. It is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i. The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all complex numbers is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> (<a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a>) or C (upright bold). In some disciplines such as <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> and <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, j is used instead of i, as i frequently represents <a href="/wiki/Electric_current" title="Electric current">electric current</a>,<a href="#cite_note-Campbell_1911-9">[8]</a><a href="#cite_note-Brown-Churchill_1996-10">[9]</a> and complex numbers are written as a + bj or a + jb. <div class="mw-heading mw-heading3"><h3 id="Addition_and_subtraction">Addition and subtraction</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=2" title="Edit section: Addition and subtraction">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_Addition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/250px-Vector_Addition.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/330px-Vector_Addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/440px-Vector_Addition.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Addition of two complex numbers can be done geometrically by constructing a parallelogram.</figcaption></figure> Two complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=x+yi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849881914f6464bb3c697317311c77a500f6132a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.456ex; height:2.509ex;" alt="{\displaystyle a=x+yi}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=u+vi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=u+vi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124f0619fed742fa17b7b1c1574f35eb2eb47a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.196ex; height:2.343ex;" alt="{\displaystyle b=u+vi}" /> are <a href="/wiki/Addition" title="Addition">added</a> by separately adding their real and imaginary parts. That is to say: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff4909ffe2aef058a635cfd6eb732d37cf26e9a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.483ex; height:2.843ex;" alt="{\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}" /> Similarly, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a> can be performed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411c471be9fd3890db6a81991fda5e3a16165961" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.483ex; height:2.843ex;" alt="{\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.}" /> The addition can be geometrically visualized as follows: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the <a href="/wiki/Triangle" title="Triangle">triangles</a> OAB and XBA are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>. <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=3" title="Edit section: Multiplication">edit</a>]</div> The product of two complex numbers is computed as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6f9b21903da13a2ad8a091b391b8ef0d279e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.812ex; height:2.843ex;" alt="{\displaystyle (a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.}" /></dd></dl> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>i</mi> <mo>=</mo> <mn>14</mn> <mo>+</mo> <mn>5</mn> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7349b1ad1793841cd53f130c69a4ae56e51fc589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.559ex; height:2.843ex;" alt="{\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.}" /> In particular, this includes as a special case the fundamental formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=i\cdot i=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mo>⋅</mo> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=i\cdot i=-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069c3dd493e19b20ac7880df02ad351dfdcc4251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.955ex; height:2.843ex;" alt="{\displaystyle i^{2}=i\cdot i=-1.}" /></dd></dl> This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number. With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the <a href="/wiki/Distributive_property" title="Distributive property">distributive property</a>, the <a href="/wiki/Commutative_property" title="Commutative property">commutative properties</a> (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, the same way as the rational or real numbers do.<a href="#cite_note-FOOTNOTEApostol198115–16-11">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Complex_conjugate,_absolute_value,_argument_and_division">Complex conjugate, absolute value, argument and division</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=4" title="Edit section: Complex conjugate, absolute value, argument and division">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_conjugate_picture.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/180px-Complex_conjugate_picture.svg.png" decoding="async" width="180" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/270px-Complex_conjugate_picture.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/360px-Complex_conjugate_picture.svg.png 2x" data-file-width="300" data-file-height="422" /></a><figcaption>Geometric representation of z and its conjugate z in the complex plane.</figcaption></figure> The <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the complex number z = x + yi is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}=x-yi.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}=x-yi.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e82e407a474c3a9436d35a7e73ed4b978005f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.081ex; height:2.676ex;" alt="{\displaystyle {\overline {z}}=x-yi.}" /><a href="#cite_note-12">[11]</a> It is also denoted by some authors by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b376dccffe5ae946dcdb7e98bf41beae28dc9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle z^{*}}" />. Geometrically, z is the <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">"reflection"</a> of z about the real axis. Conjugating twice gives the original complex number: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\overline {z}}}=z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\overline {z}}}=z.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c1dd54f680ab88e888796ab01626eaa79c5035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.156ex; height:3.176ex;" alt="{\displaystyle {\overline {\overline {z}}}=z.}" /> A complex number is real if and only if it equals its own conjugate. The <a href="/wiki/Unary_operation" title="Unary operation">unary operation</a> of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_number_illustration_modarg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Complex_number_illustration_modarg.svg/220px-Complex_number_illustration_modarg.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Complex_number_illustration_modarg.svg/330px-Complex_number_illustration_modarg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Complex_number_illustration_modarg.svg/440px-Complex_number_illustration_modarg.svg.png 2x" data-file-width="180" data-file-height="180" /></a><figcaption>Argument φ and modulus r locate a point in the complex plane.</figcaption></figure> For any complex number z = x + yi , the product <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44170c8ad144e96fda11d9c39fb5d706b39b2b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.486ex; height:3.176ex;" alt="{\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}" /></dd></dl> is a non-negative real number. This allows to define the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> (or modulus or magnitude) of z to be the <a href="/wiki/Square_root" title="Square root">square root</a><a href="#cite_note-FOOTNOTEApostol198118-13">[12]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bd44a6d60e8a02c0646ab894fd7b9743eab576" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.89ex; height:4.843ex;" alt="{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}" /> By <a href="/wiki/Pythagoras%27_theorem" class="mw-redirect" title="Pythagoras' theorem">Pythagoras' theorem</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28fd4d7dcabf618d707c21bd08306c7b3aa8b68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.382ex; height:2.843ex;" alt="{\displaystyle |z|}" /> is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the <a href="/wiki/Unit_circle" title="Unit circle">circle of radius one</a> around the origin consists precisely of the numbers z such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3749e5cd50ee274eb73aea2ade8441687140a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|=1}" />. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x=x+0i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x=x+0i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d28262d6a55ec5c2d7bc25e8bf59c15a36417f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.75ex; height:2.343ex;" alt="{\displaystyle z=x=x+0i}" /> is a real number, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=|x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=|x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e312c629a7e0345d2fa692c14b90e246f4548e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.104ex; height:2.843ex;" alt="{\displaystyle |z|=|x|}" />: its absolute value as a complex number and as a real number are equal. Using the conjugate, the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of a nonzero complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+yi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+yi}" /> can be computed to be <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mi>i</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8259d41cc02921a69f715e57345d301979e85f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}" /> More generally, the division of an arbitrary complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=u+vi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=u+vi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbb3bc012ec43f3f2e0781f99b45e292f9c98be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.863ex; height:2.343ex;" alt="{\displaystyle w=u+vi}" /> by a non-zero complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+yi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+yi}" /> equals <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mi>y</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d152defafb391f742352a9f996c96533bca8c27a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.278ex; height:6.843ex;" alt="{\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.}" /> This process is sometimes called "<a href="/wiki/Rationalisation_(mathematics)" title="Rationalisation (mathematics)">rationalization</a>" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.<a href="#cite_note-14">[13]</a><a href="#cite_note-15">[14]</a> The <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of z (sometimes called the "phase" φ)<a href="#cite_note-:2-7">[7]</a> is the angle of the <a href="/wiki/Radius" title="Radius">radius</a> Oz with the positive real axis, and is written as arg z, expressed in <a href="/wiki/Radian" title="Radian">radians</a> in this article. The angle is defined only up to adding integer multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" />, since a rotation by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" /> (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb1843079a9df3d3bbcce3249bb2599790de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ]}" />, which is referred to as the <a href="/wiki/Principal_value" title="Principal value">principal value</a>.<a href="#cite_note-16">[15]</a> The argument can be computed from the rectangular form x + yi by means of the <a href="/wiki/Arctan" class="mw-redirect" title="Arctan">arctan</a> (inverse tangent) function.<a href="#cite_note-17">[16]</a> <div class="mw-heading mw-heading3"><h3 id="Polar_form">Polar form</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=5" title="Edit section: Polar form">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">Polar coordinate system</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"Polar form" redirects here. For the higher-dimensional analogue, see <a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_multi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Complex_multi.svg/250px-Complex_multi.svg.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Complex_multi.svg/330px-Complex_multi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Complex_multi.svg/500px-Complex_multi.svg.png 2x" data-file-width="700" data-file-height="600" /></a><figcaption>Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms φ1+φ2 in the equation) and stretched by the length of the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of the blue triangle (the multiplication of both radiuses, as per term r1r2 in the equation).</figcaption></figure> For any complex number z, with absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=|z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=|z|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd846958260995e3ecc934b403748988a49e9511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.529ex; height:2.843ex;" alt="{\displaystyle r=|z|}" /> and argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" />, the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r(\cos \varphi +i\sin \varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r(\cos \varphi +i\sin \varphi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe097f200e7ea38fe974bf69e6af9a50711f431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.856ex; height:2.843ex;" alt="{\displaystyle z=r(\cos \varphi +i\sin \varphi )}" /></dd></dl> holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z=r\operatorname {\mathrm {cis} } \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z=r\operatorname {\mathrm {cis} } \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac47d378cacc9cdc321ea3aaa6e174f90afc237b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.125ex; height:2.676ex;" alt="{\textstyle z=r\operatorname {\mathrm {cis} } \varphi }" />. In <a href="/wiki/Electronics" title="Electronics">electronics</a>, one represents a <a href="/wiki/Phasor_(sine_waves)" class="mw-redirect" title="Phasor (sine waves)">phasor</a> with amplitude r and phase φ in <a href="/wiki/Angle_notation" class="mw-redirect" title="Angle notation">angle notation</a>:<a href="#cite_note-18">[17]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r\angle \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mi mathvariant="normal">∠</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r\angle \varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b49277ca2aa60836f3415a9a26cfab749b0b07c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.08ex; height:2.676ex;" alt="{\displaystyle z=r\angle \varphi .}" /> If two complex numbers are given in polar form, i.e., z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), the product and division can be computed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6502c352808cfc910a170a23813f02822e9b758" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.625ex; height:2.843ex;" alt="{\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b0cccea4ebf09b067273a74240582a313ac66c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:51.203ex; height:5.009ex;" alt="{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.}" /> (These are a consequence of the <a href="/wiki/Trigonometric_identities" class="mw-redirect" title="Trigonometric identities">trigonometric identities</a> for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2+i)(3+i)=5+5i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mn>5</mn> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+i)(3+i)=5+5i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38aeed692cd66f9df75aebfa019e3d57aeeb56b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.942ex; height:2.843ex;" alt="{\displaystyle (2+i)(3+i)=5+5i.}" /> Because the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in <a href="/wiki/Radian" title="Radian">radian</a>). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are <a href="/wiki/Arctan" class="mw-redirect" title="Arctan">arctan</a>(1/3) and arctan(1/2), respectively. Thus, the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cb184ba7ee6d5c7f21a3cf8e8c893cb2e997bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.879ex; height:6.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}" /> holds. As the <a href="/wiki/Arctan" class="mw-redirect" title="Arctan">arctan</a> function can be approximated highly efficiently, formulas like this – known as <a href="/wiki/Machin-like_formula" title="Machin-like formula">Machin-like formulas</a> – are used for high-precision approximations of <a href="/wiki/Pi" title="Pi">π</a>:<a href="#cite_note-19">[18]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=4\arctan \left({\frac {1}{5}}\right)-\arctan \left({\frac {1}{239}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>239</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=4\arctan \left({\frac {1}{5}}\right)-\arctan \left({\frac {1}{239}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cfe34b34fdaa317b034cb06549651e0db72438" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.753ex; height:6.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=4\arctan \left({\frac {1}{5}}\right)-\arctan \left({\frac {1}{239}}\right)}" /> <div class="mw-heading mw-heading3"><h3 id="Powers_and_roots">Powers and roots</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=6" title="Edit section: Powers and roots">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Square_root#Square_roots_of_negative_and_complex_numbers" title="Square root">Square roots of negative and complex numbers</a></div> The n-th power of a complex number can be computed using <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a>, which is obtained by repeatedly applying the above formula for the product: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>z</mi> <mo>⋅</mo> <mo>⋯</mo> <mo>⋅</mo> <mi>z</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> factors</mtext> </mrow> </mrow> </munder> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eafeb1be171acd8a945e5c9d9046abe4f35ad5b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:61.27ex; height:5.843ex;" alt="{\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}" /> For example, the first few powers of the imaginary unit i are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75483975591d7804e753c30d1c9e3a59295dd599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.062ex; height:3.009ex;" alt="{\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }" />. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visualisation_complex_number_roots.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/330px-Visualisation_complex_number_roots.svg.png" decoding="async" width="300" height="400" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/500px-Visualisation_complex_number_roots.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/600px-Visualisation_complex_number_roots.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>Geometric representation of the 2nd to 6th roots of a complex number z, in polar form reiφ  where r = |z | and φ = arg z. If z is real, φ = 0 or π. Principal roots are shown in black.</figcaption></figure> The n <a href="/wiki/Nth_root" title="Nth root">nth roots</a> of a complex number z are given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc1b3406644f788c1ac1799d6328118ee66516f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.488ex; height:6.176ex;" alt="{\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}" /> for 0 ≤ k ≤ n − 1. (Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10eb7386bd8efe4c5b5beafe05848fbd923e1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{r}}}" /> is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values. For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8b7eb2d2a30057811a7835502717d3d6ece962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}" />, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}=1,z_{2}=i,z_{3}=-1,z_{4}=-i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}=1,z_{2}=i,z_{3}=-1,z_{4}=-i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b460707ba2916805ea7ce8a4212d1db2749e27ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.23ex; height:2.509ex;" alt="{\displaystyle z_{1}=1,z_{2}=i,z_{3}=-1,z_{4}=-i.}" /></dd></dl> In general there is no natural way of distinguishing one particular complex nth root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n-th root, which is therefore commonly referred to as the n-th root of x.) One refers to this situation by saying that the nth root is a <a href="/wiki/Multivalued_function" title="Multivalued function">n-valued function</a> of z. <div class="mw-heading mw-heading3"><h3 id="Fundamental_theorem_of_algebra">Fundamental theorem of algebra</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=7" title="Edit section: Fundamental theorem of algebra">edit</a>]</div> The <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>, of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> and <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a>, states that for any complex numbers (called <a href="/wiki/Coefficient" title="Coefficient">coefficients</a>) a0, ..., an, the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd18e443bc73d8469f68a8a4d62e4abd5a5c162f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.919ex; height:2.676ex;" alt="{\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}" /> has at least one complex solution z, provided that at least one of the higher coefficients a1, ..., an is nonzero.<a href="#cite_note-Bourbaki_1998_loc=§VIII.1-20">[19]</a> This property does not hold for the <a href="/wiki/Rational_number" title="Rational number">field of rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> (the polynomial x2 − 2 does not have a rational root, because <a href="/wiki/Square_root_of_2" title="Square root of 2">√2</a> is not a rational number) nor the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> (the polynomial x2 + 4 does not have a real root, because the square of x is positive for any real number x). Because of this fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is called an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>. It is a cornerstone of various applications of complex numbers, as is detailed further below. There are various proofs of this theorem, by either analytic methods such as <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>, or <a href="/wiki/Topology" title="Topology">topological</a> ones such as the <a href="/wiki/Winding_number" title="Winding number">winding number</a>, or a proof combining <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> and the fact that any real polynomial of odd degree has at least one real root. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=8" title="Edit section: History">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Negative_number#History" title="Negative number">Negative number § History</a></div> The solution in <a href="/wiki/Nth_root" title="Nth root">radicals</a> (without <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>) of a general <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equation</a>, when all three of its roots are real numbers, contains the square roots of <a href="/wiki/Negative_numbers" class="mw-redirect" title="Negative numbers">negative numbers</a>, a situation that cannot be rectified by factoring aided by the <a href="/wiki/Rational_root_test" class="mw-redirect" title="Rational root test">rational root test</a>, if the cubic is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a>; this is the so-called <a href="/wiki/Casus_irreducibilis" title="Casus irreducibilis">casus irreducibilis</a> ("irreducible case"). This conundrum led Italian mathematician <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> to conceive of complex numbers in around 1545 in his <a href="/wiki/Ars_Magna_(Cardano_book)" title="Ars Magna (Cardano book)">Ars Magna</a>,<a href="#cite_note-21">[20]</a> though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".<a href="#cite_note-22">[21]</a> Cardano did use imaginary numbers, but described using them as "mental torture."<a href="#cite_note-23">[22]</a> This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably <a href="/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a>, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.<a href="#cite_note-24">[23]</a> Work on the problem of general polynomials ultimately led to the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>, which shows that with complex numbers, a solution exists to every <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> of degree one or higher. Complex numbers thus form an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>, where any polynomial equation has a <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a>. Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician <a href="/wiki/Rafael_Bombelli" title="Rafael Bombelli">Rafael Bombelli</a>.<a href="#cite_note-25">[24]</a> A more abstract formalism for the complex numbers was further developed by the Irish mathematician <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>, who extended this abstraction to the theory of <a href="/wiki/Quaternions" class="mw-redirect" title="Quaternions">quaternions</a>.<a href="#cite_note-26">[25]</a> The earliest fleeting reference to <a href="/wiki/Square_root" title="Square root">square roots</a> of <a href="/wiki/Negative_number" title="Negative number">negative numbers</a> can perhaps be said to occur in the work of the <a href="/wiki/Hellenistic_mathematics" class="mw-redirect" title="Hellenistic mathematics">Greek mathematician</a> <a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Hero of Alexandria</a> in the 1st century <a href="/wiki/AD" class="mw-redirect" title="AD">AD</a>, where in his <a href="/wiki/Hero_of_Alexandria#Bibliography" title="Hero of Alexandria">Stereometrica</a> he considered, apparently in error, the volume of an impossible <a href="/wiki/Frustum" title="Frustum">frustum</a> of a <a href="/wiki/Pyramid" title="Pyramid">pyramid</a> to arrive at the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {81-144}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>81</mn> <mo>−</mo> <mn>144</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {81-144}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e362fba3f817d73fb17a47ab312f478bde84773c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.589ex; height:3.009ex;" alt="{\displaystyle {\sqrt {81-144}}}" /> in his calculations, which today would simplify to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>63</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accaf396b68458754b5cbe532bf7a3160f3acb78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.231ex; height:3.176ex;" alt="{\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}}" />.<a href="#cite_note-28">[b]</a> Negative quantities were not conceived of in <a href="/wiki/Hellenistic_mathematics" class="mw-redirect" title="Hellenistic mathematics">Hellenistic mathematics</a> and Hero merely replaced the negative value by its positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>144</mn> <mo>−</mo> <mn>81</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc15d48ccea867a37beef8358473f0c240dddf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.595ex; height:3.176ex;" alt="{\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.}" /><a href="#cite_note-29">[27]</a> The impetus to study complex numbers as a topic in itself first arose in the 16th century when <a href="/wiki/Algebraic_solution" class="mw-redirect" title="Algebraic solution">algebraic solutions</a> for the roots of <a href="/wiki/Cubic_equation" title="Cubic equation">cubic</a> and <a href="/wiki/Quartic_equation" title="Quartic equation">quartic</a> <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> were discovered by Italian mathematicians (<a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> and <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>). It was soon realized (but proved much later)<a href="#cite_note-Casus-30">[28]</a> that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers <a href="/wiki/Casus_irreducibilis" title="Casus irreducibilis">is unavoidable</a> when all three roots are real and distinct.<a href="#cite_note-31">[c]</a> However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues. The term "imaginary" for these quantities was coined by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> in 1637, who was at pains to stress their unreal nature:<a href="#cite_note-32">[29]</a> <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">... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine. [... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.]</blockquote> A further source of confusion was that the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01130abdb35d388ef63d1484ac51a33dc02aec1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.941ex; height:3.509ex;" alt="{\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}" /> seemed to be capriciously inconsistent with the algebraic identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>b</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a6fe99883dd2ee2bda43eab716e18d9bece3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.361ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}" />, which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>a</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc8d02f310ed2784e426bda06a22b24c278275e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.203ex; height:5.009ex;" alt="{\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}" />, even bedeviled <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. This difficulty eventually led to the convention of using the special symbol i in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" /> to guard against this mistake.<a href="#cite_note-33">[30]</a><a href="#cite_note-34">[31]</a> Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, <a href="/wiki/Elements_of_Algebra" title="Elements of Algebra">Elements of Algebra</a>, he introduces these numbers almost at once and then uses them in a natural way throughout. In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c100a9d6c15a5c191d1de4330644da02c4bc7ee4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.466ex; height:2.843ex;" alt="{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .}" /> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_cos_sin.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/330px-Circle_cos_sin.gif" decoding="async" width="330" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Circle_cos_sin.gif/500px-Circle_cos_sin.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3b/Circle_cos_sin.gif 2x" data-file-width="650" data-file-height="390" /></a><figcaption>Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing <a href="/wiki/Uniform_circular_motion" class="mw-redirect" title="Uniform circular motion">uniform circular motion</a> in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.</figcaption></figure> In 1748, Euler went further and obtained <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>:<a href="#cite_note-35">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49bcb7ddc21b4c2d70983137c061fe72b9171719" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.704ex; height:2.843ex;" alt="{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }" /> by formally manipulating complex <a href="/wiki/Power_series" title="Power series">power series</a> and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. The idea of a complex number as a point in the complex plane (<a href="#Complex_plane">above</a>) was first described by <a href="/wiki/Denmark" title="Denmark">Danish</a>–<a href="/wiki/Norway" title="Norway">Norwegian</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> <a href="/wiki/Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a> in 1799,<a href="#cite_note-36">[33]</a> although it had been anticipated as early as 1685 in <a href="/wiki/John_Wallis" title="John Wallis">Wallis's</a> A Treatise of Algebra.<a href="#cite_note-37">[34]</a> Wessel's memoir appeared in the Proceedings of the <a href="/wiki/Copenhagen_Academy" class="mw-redirect" title="Copenhagen Academy">Copenhagen Academy</a> but went largely unnoticed. In 1806 <a href="/wiki/Jean-Robert_Argand" title="Jean-Robert Argand">Jean-Robert Argand</a> independently issued a pamphlet on complex numbers and provided a rigorous proof of the <a href="/wiki/Fundamental_theorem_of_algebra#History" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>.<a href="#cite_note-38">[35]</a> <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> had earlier published an essentially <a href="/wiki/Topology" title="Topology">topological</a> proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".<a href="#cite_note-39">[36]</a> It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,<a href="#cite_note-Ewald-40">[37]</a> largely establishing modern notation and terminology:<a href="#cite_note-FOOTNOTEGauss1831-41">[38]</a> <blockquote>If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" /> positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.</blockquote> In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,<a href="#cite_note-42">[39]</a><a href="#cite_note-43">[40]</a> <a href="/wiki/C._V._Mourey" title="C. V. Mourey">Mourey</a>,<a href="#cite_note-44">[41]</a> <a href="/w/index.php?title=John_Warren_(mathematician)&action=edit&redlink=1" class="new" title="John Warren (mathematician) (page does not exist)">Warren</a>,<a href="#cite_note-45">[42]</a><a href="#cite_note-46">[43]</a><a href="#cite_note-47">[44]</a> <a href="/wiki/Jacques_Fr%C3%A9d%C3%A9ric_Fran%C3%A7ais" title="Jacques Frédéric Français">Français</a> and his brother, <a href="/wiki/Giusto_Bellavitis" title="Giusto Bellavitis">Bellavitis</a>.<a href="#cite_note-48">[45]</a><a href="#cite_note-49">[46]</a> The English mathematician <a href="/wiki/G.H._Hardy" class="mw-redirect" title="G.H. Hardy">G.H. Hardy</a> remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as <a href="/wiki/Norway" title="Norway">Norwegian</a> <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a> and <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a> were necessarily using them routinely before Gauss published his 1831 treatise.<a href="#cite_note-50">[47]</a> <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> and <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> together brought the fundamental ideas of <a href="#Complex_analysis">complex analysis</a> to a high state of completion, commencing around 1825 in Cauchy's case. The common terms used in the theory are chiefly due to the founders. Argand called cos φ + i sin φ the direction factor, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06427751d7f71ba70ddfae47fb47e6386324ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.647ex; height:3.509ex;" alt="{\displaystyle r={\sqrt {a^{2}+b^{2}}}}" /> the modulus;<a href="#cite_note-51">[d]</a><a href="#cite_note-52">[48]</a> Cauchy (1821) called cos φ + i sin φ the reduced form (l'expression réduite)<a href="#cite_note-53">[49]</a> and apparently introduced the term argument; Gauss used i for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" />,<a href="#cite_note-55">[e]</a> introduced the term complex number for a + bi,<a href="#cite_note-57">[f]</a> and called a2 + b2 the norm.<a href="#cite_note-59">[g]</a> The expression direction coefficient, often used for cos φ + i sin φ, is due to Hankel (1867),<a href="#cite_note-60">[53]</a> and absolute value, for modulus, is due to Weierstrass. Later classical writers on the general theory include <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>, <a href="/wiki/Otto_H%C3%B6lder" title="Otto Hölder">Otto Hölder</a>, <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>, <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, <a href="/wiki/Hermann_Schwarz" title="Hermann Schwarz">Hermann Schwarz</a>, <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by <a href="/wiki/Wilhelm_Wirtinger" title="Wilhelm Wirtinger">Wilhelm Wirtinger</a> in 1927. <div class="mw-heading mw-heading2"><h2 id="Abstract_algebraic_aspects">Abstract algebraic aspects</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=9" title="Edit section: Abstract algebraic aspects">edit</a>]</div> While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately. <div class="mw-heading mw-heading3"><h3 id="Construction_as_a_quotient_field">Construction as a quotient field</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=10" title="Edit section: Construction as a quotient field">edit</a>]</div> One approach to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is via <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, i.e., expressions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec820b19602babe3261421d56db1d4023327d517" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:32.25ex; height:2.843ex;" alt="{\displaystyle p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}" /> where the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> a0, ..., an are real numbers. The set of all such polynomials is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}" />. Since sums and products of polynomials are again polynomials, this set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}" /> forms a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, called the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> (over the reals). To every such polynomial p, one may assign the complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6ad564b89563b2a749f6ecc7afb9cbfc2c03bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:28.053ex; height:2.843ex;" alt="{\displaystyle p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}}" />, i.e., the value obtained by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b73cd07a74c26124a71211e820932d3c3db9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.881ex; height:2.176ex;" alt="{\displaystyle X=i}" />. This defines a function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [X]\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]\to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b404ca14e700bc8fd42f11a126173d5c1a6cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]\to \mathbb {C} }" /></dd></dl> This function is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> since every complex number can be obtained in such a way: the evaluation of a <a href="/wiki/Linear_polynomial" class="mw-redirect" title="Linear polynomial">linear polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bX}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5038aa69711746adfd10287ec835eb585a036ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.048ex; height:2.343ex;" alt="{\displaystyle a+bX}" /> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b73cd07a74c26124a71211e820932d3c3db9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.881ex; height:2.176ex;" alt="{\displaystyle X=i}" /> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}" />. However, the evaluation of polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759c679330a1c67db74a3da9ee5cca488de3a589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle X^{2}+1}" /> at i is 0, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}+1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}+1=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37bfb6199846fe6d16ecbb7be96c5ca3848fdcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.767ex; height:2.843ex;" alt="{\displaystyle i^{2}+1=0.}" /> This polynomial is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a>, i.e., cannot be written as a product of two linear polynomials. Basic facts of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> then imply that the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of the above map is an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by this polynomial, and that the quotient by this ideal is a field, and that there is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≅</mo> </mrow> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a397538266a79eecf6b7e746fb7791a3bcf532a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.979ex; height:3.843ex;" alt="{\displaystyle \mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} }" /></dd></dl> between the quotient ring and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. Some authors take this as the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />.<a href="#cite_note-61">[54]</a> Accepting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is algebraically closed, because it is an <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> in this approach, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is therefore the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" /> <div class="mw-heading mw-heading3"><h3 id="Matrix_representation_of_complex_numbers">Matrix representation of complex numbers</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=11" title="Edit section: Matrix representation of complex numbers">edit</a>]</div> Complex numbers a + bi can also be represented by 2 × 2 <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> that have the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5f0db84bd94b46060f6d631fdda4a7b65f2da7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.178ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.}" /> Here the entries a and b are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a <a href="/wiki/Subring" title="Subring">subring</a> of the ring of 2 × 2 matrices. A simple computation shows that the map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb2a42d93e6c0c6dc4fd84d2c534d1ccd736bf1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.015ex; height:6.176ex;" alt="{\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}" /> is a <a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">ring isomorphism</a> from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the corresponding matrix, and the conjugate of a complex number with the <a href="/wiki/Transpose" title="Transpose">transpose</a> of the matrix. The geometric description of the multiplication of complex numbers can also be expressed in terms of <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrices</a> by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector (x, y) corresponds to the multiplication of x + iy by a + ib. In particular, if the determinant is 1, there is a real number t such that the matrix has the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mtd> <mtd> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fae36b2b1b5e42dbc6cb42faff690b21bb9af58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.757ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.}" /> In this case, the action of the matrix on vectors and the multiplication by the complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos t+i\sin t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos t+i\sin t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440a8a3358e2ea3203a610d0a76aad876bd1cbe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.45ex; height:2.343ex;" alt="{\displaystyle \cos t+i\sin t}" /> are both the <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> of the angle t. <div class="mw-heading mw-heading2"><h2 id="Complex_analysis">Complex analysis</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=12" title="Edit section: Complex analysis">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></div> The study of functions of a complex variable is known as <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> and has enormous practical use in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> as well as in other branches of mathematics. Often, the most natural proofs for statements in <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> or even <a href="/wiki/Number_theory" title="Number theory">number theory</a> employ techniques from complex analysis (see <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> for an example). <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex-plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/220px-Complex-plot.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/330px-Complex-plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/440px-Complex-plot.png 2x" data-file-width="579" data-file-height="445" /></a><figcaption>A <a href="/wiki/Domain_coloring" title="Domain coloring">domain coloring</a> graph of the function <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠(z2 − 1)(z − 2 − i)2/z2 + 2 + 2i⁠. Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for ±1, (2 + i) and <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {-2-2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mi>i</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {-2-2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bc7e5f92a9bc9585b7db872d44fd3cb7fb9665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.167ex; height:3.009ex;" alt="{\displaystyle \pm {\sqrt {-2-2i}}.}" /></figcaption></figure> Unlike real functions, which are commonly represented as two-dimensional graphs, <a href="/wiki/Complex_function" class="mw-redirect" title="Complex function">complex functions</a> have four-dimensional graphs and may usefully be illustrated by color-coding a <a href="/wiki/Graph_of_a_function_of_two_variables" class="mw-redirect" title="Graph of a function of two variables">three-dimensional graph</a> to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane. <div class="mw-heading mw-heading3"><h3 id="Convergence">Convergence</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=13" title="Edit section: Convergence">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ComplexPowers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ComplexPowers.svg/220px-ComplexPowers.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ComplexPowers.svg/330px-ComplexPowers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ComplexPowers.svg/440px-ComplexPowers.svg.png 2x" data-file-width="354" data-file-height="354" /></a><figcaption>Illustration of the behavior of the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1a8cdd7ee39054e510deeb38ee551cc7616ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}" /> for three different values of z (all having the same argument): for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c0fa57b899b653a3823f85f43fd666309c09b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|<1}" /> the sequence converges to 0 (inner spiral), while it diverges for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b679aa1ea7b5c6d6d06a1210b4923aad2c017377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|>1}" /> (outer spiral).</figcaption></figure> The notions of <a href="/wiki/Convergent_series" title="Convergent series">convergent series</a> and <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to <a href="/wiki/Convergent_sequence" class="mw-redirect" title="Convergent sequence">converge</a> if and only if its real and imaginary parts do. This is equivalent to the <a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limits</a>, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />, endowed with the <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bd602f9ebc09f350085c4805dea85646a4c120" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.91ex; height:2.843ex;" alt="{\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|}" /> is a complete <a href="/wiki/Metric_space" title="Metric space">metric space</a>, which notably includes the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2999b413c874f0ee618486154b679ef6875d48c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.202ex; height:2.843ex;" alt="{\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|}" /> for any two complex numbers z1 and z2. <div class="mw-heading mw-heading3"><h3 id="Complex_exponential">Complex exponential</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=14" title="Edit section: Complex exponential">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ComplexExpMapping.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/ComplexExpMapping.svg/220px-ComplexExpMapping.svg.png" decoding="async" width="220" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/ComplexExpMapping.svg/330px-ComplexExpMapping.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/ComplexExpMapping.svg/440px-ComplexExpMapping.svg.png 2x" data-file-width="638" data-file-height="306" /></a><figcaption>Illustration of the complex exponential function mapping the complex plane, w = exp ⁡(z). The left plane shows a square mesh with mesh size 1, with the three complex numbers 0, 1, and i highlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to the x-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to the y-axis are mapped to circles.</figcaption></figure> Like in real analysis, this notion of convergence is used to construct a number of <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a>: the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> exp z, also written ez, is defined as the <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>, which can be shown to <a href="/wiki/Radius_of_convergence" title="Radius of convergence">converge</a> for any z: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>z</mi> <mo>:=</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8ea97a6ca2dd64faf189a995c6cc80af1cde86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.263ex; height:6.843ex;" alt="{\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}" /> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bcf5271c78981c0ca2f2ca46b841621b1c284e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.524ex; height:2.843ex;" alt="{\displaystyle \exp(1)}" /> is <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\approx 2.718}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>≈</mo> <mn>2.718</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\approx 2.718}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e2bc9d17c0545d9f2792476c5473f296957270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.479ex; height:2.176ex;" alt="{\displaystyle e\approx 2.718}" />. <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> states: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1578d4fc73aca4efba684f9c66a218c6c871a32a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.594ex; height:2.843ex;" alt="{\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi }" /> for any real number φ. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes <a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(i\pi )=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(i\pi )=-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089533cfe83d130a1c07429923de0259762830d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.212ex; height:2.843ex;" alt="{\displaystyle \exp(i\pi )=-1.}" /> <div class="mw-heading mw-heading3"><h3 id="Complex_logarithm">Complex logarithm</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=15" title="Edit section: Complex logarithm">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ComplexExpStrips.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/ComplexExpStrips.svg/220px-ComplexExpStrips.svg.png" decoding="async" width="220" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/ComplexExpStrips.svg/330px-ComplexExpStrips.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/ComplexExpStrips.svg/440px-ComplexExpStrips.svg.png 2x" data-file-width="638" data-file-height="306" /></a><figcaption>The exponential function maps complex numbers z differing by a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5715af49984c5b33961d55f532d14497b0cbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.297ex; height:2.176ex;" alt="{\displaystyle 2\pi i}" /> to the same complex number w.</figcaption></figure> For any positive real number t, there is a unique real number x such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)=t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(x)=t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b61ef91b2e17b0e5ab7bc44ff2dbb389557353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.629ex; height:2.843ex;" alt="{\displaystyle \exp(x)=t}" />. This leads to the definition of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> as the <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>;</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd4ef60a8d8dd5a7db33ec3e1380a38912ebb29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.088ex; height:2.843ex;" alt="{\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x}" /> of the exponential function. The situation is different for complex numbers, since <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mi>z</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25dd3b4b438eb2e7d90e7ae6f586f00a54e36a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.885ex; height:2.843ex;" alt="{\displaystyle \exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z}" /></dd></dl> by the functional equation and Euler's identity. For example, eiπ = e3iπ = −1 , so both iπ and 3iπ are possible values for the complex logarithm of −1. In general, given any non-zero complex number w, any number z solving the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp z=w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp z=w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6232a6e2d35e890d2443d98ff102ad17404326e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.79ex; height:2.009ex;" alt="{\displaystyle \exp z=w}" /></dd></dl> is called a <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> of w, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23246cf4aef2e1c068cd85c66b4ebf1a6c56320a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.023ex; height:2.509ex;" alt="{\displaystyle \log w}" />. It can be shown that these numbers satisfy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\log w=\ln |w|+i\arg w,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mi>w</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>arg</mi> <mo>⁡</mo> <mi>w</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\log w=\ln |w|+i\arg w,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bbc16095e9164a51da571251a59b3f77e2b43cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.557ex; height:2.843ex;" alt="{\displaystyle z=\log w=\ln |w|+i\arg w,}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec03a9c123925f400a40064ca491d268f9312956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.009ex;" alt="{\displaystyle \arg }" /> is the <a href="/wiki/Arg_(mathematics)" class="mw-redirect" title="Arg (mathematics)">argument</a> defined <a href="#Polar_form">above</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0de5ba4f372ede555d00035e70c50ed0b9625d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \ln }" /> the (real) <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. As arg is a <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued function</a>, unique only up to a multiple of 2π, log is also multivalued. The <a href="/wiki/Principal_value" title="Principal value">principal value</a> of log is often taken by restricting the imaginary part to the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> (−π, π]. This leads to the complex logarithm being a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> function taking values in the strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mspace width="thickmathspace"></mspace> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d836fb007d819a1aab60ece11449d6d754192c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.309ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right]}" /> (that is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" /> in the above illustration) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>:</mo> <mspace width="thickmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo stretchy="false">→</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <mspace width="thickmathspace"></mspace> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9195ba0433fd0b1768386d0e3b2c11fb5eb684" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.345ex; height:3.009ex;" alt="{\displaystyle \ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right].}" /> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo class="MJX-variant">∖</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d740d61e0afa8776c8081f366c9d94c612620b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.43ex; height:2.843ex;" alt="{\displaystyle z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)}" /> is not a non-positive real number (a positive or a non-real number), the resulting <a href="/wiki/Principal_value" title="Principal value">principal value</a> of the complex logarithm is obtained with −π < φ < π. It is an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in -\mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in -\mathbb {R} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f55f57d48954b4f712e2550445ee066490d74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.926ex; height:2.676ex;" alt="{\displaystyle z\in -\mathbb {R} ^{+}}" />, where the principal value is ln z = ln(−z) + iπ.<a href="#cite_note-62">[h]</a> Complex <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> zω is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{\omega }=\exp(\omega \ln z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{\omega }=\exp(\omega \ln z),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e625fe27ba8c070e5376bb0e92c44fa5d4bc244" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.699ex; height:2.843ex;" alt="{\displaystyle z^{\omega }=\exp(\omega \ln z),}" /> and is multi-valued, except when ω is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above. If z > 0 is real (and ω an arbitrary complex number), one has a preferred choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}" />, the real logarithm, which can be used to define a preferred exponential function. Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see <a href="/wiki/Exponentiation#Failure_of_power_and_logarithm_identities" title="Exponentiation">failure of power and logarithm identities</a>. For example, they do not satisfy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00aee0bc32a306ac68a1521f059c934e48611371" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.866ex; height:3.509ex;" alt="{\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}" /> Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right. <div class="mw-heading mw-heading3"><h3 id="Complex_sine_and_cosine">Complex sine and cosine</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=16" title="Edit section: Complex sine and cosine">edit</a>]</div> The series defining the real trigonometric functions <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, as well as the <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a> sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as <a href="/wiki/Tangent_(function)" class="mw-redirect" title="Tangent (function)">tangent</a>, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>. <div class="mw-heading mw-heading3"><h3 id="Holomorphic_functions">Holomorphic functions</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=17" title="Edit section: Holomorphic functions">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sin1z-cplot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sin1z-cplot.svg/220px-Sin1z-cplot.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sin1z-cplot.svg/330px-Sin1z-cplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sin1z-cplot.svg/440px-Sin1z-cplot.svg.png 2x" data-file-width="565" data-file-height="426" /></a><figcaption>Color wheel graph of the function sin(1/z) that is holomorphic except at z = 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.</figcaption></figure> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bd74de42920d73678106d48b81416d96f3aec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.894ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {C} }" /> → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is called <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> or complex differentiable at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" /> if the limit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo stretchy="false">→</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c6d93c97b63a8602179e2c96d1fdee50f488a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.472ex; height:6.009ex;" alt="{\displaystyle \lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}}" /></dd></dl> exists (in which case it is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85049970069b0d6c40718cf3dab2cf4757faae30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.95ex; height:3.009ex;" alt="{\displaystyle f'(z_{0})}" />). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" /> in different directions imposes a much stronger condition than being (real) differentiable. For example, the function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\overline {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\overline {z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75311667f3ed9db08d4f87510c37e372a2c87d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.482ex; height:2.843ex;" alt="{\displaystyle f(z)={\overline {z}}}" /></dd></dl> is differentiable as a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598df77137da45a239ab44e369e851b66a60db0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.079ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}" />, but is not complex differentiable. A real differentiable function is complex differentiable <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it satisfies the <a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>, which are sometimes abbreviated as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eea153148dfb0c706b4d4d654bfa322e2b3c0a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.341ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}=0.}" /></dd></dl> Complex analysis shows some features not apparent in real analysis. For example, the <a href="/wiki/Identity_theorem" title="Identity theorem">identity theorem</a> asserts that two holomorphic functions f and g agree if they agree on an arbitrarily small <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. <a href="/wiki/Meromorphic_function" title="Meromorphic function">Meromorphic functions</a>, functions that can locally be written as f(z)/(z − z0)n with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have <a href="/wiki/Essential_singularity" title="Essential singularity">essential singularities</a>, such as sin(1/z) at z = 0. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=18" title="Edit section: Applications">edit</a>]</div> Complex numbers have applications in many scientific areas, including <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Control_theory" title="Control theory">control theory</a>, <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Cartography" title="Cartography">cartography</a>, and <a href="/wiki/Vibration#Vibration_analysis" title="Vibration">vibration analysis</a>. Some of these applications are described below. Complex conjugation is also employed in <a href="/wiki/Inversive_geometry" title="Inversive geometry">inversive geometry</a>, a branch of geometry studying reflections more general than ones about a line. In the <a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">network analysis of electrical circuits</a>, the complex conjugate is used in finding the equivalent impedance when the <a href="/wiki/Maximum_power_transfer_theorem" title="Maximum power transfer theorem">maximum power transfer theorem</a> is looked for. <div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=19" title="Edit section: Geometry">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Shapes">Shapes</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=20" title="Edit section: Shapes">edit</a>]</div> Three <a href="/wiki/Collinearity" title="Collinearity">non-collinear</a> points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v,w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cabca98f60f9ee828adb0d73276eb90eb2ee56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.189ex; height:2.009ex;" alt="{\displaystyle u,v,w}" /> in the plane determine the <a href="/wiki/Shape#Similarity_classes" title="Shape">shape</a> of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v,w\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v,w\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b64a901f197da5658f531c5b4cbf0ec9c425265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.514ex; height:2.843ex;" alt="{\displaystyle \{u,v,w\}}" />. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1275fc01560cb752cb3f02f3da8a2087a30cd91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.913ex; height:5.176ex;" alt="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}" /> The shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>), corresponding to the intuitive notion of shape, and describing <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity</a>. Thus each triangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v,w\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v,w\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b64a901f197da5658f531c5b4cbf0ec9c425265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.514ex; height:2.843ex;" alt="{\displaystyle \{u,v,w\}}" /> is in a <a href="/wiki/Shape#Similarity_classes" title="Shape">similarity class</a> of triangles with the same shape.<a href="#cite_note-63">[55]</a> <div class="mw-heading mw-heading4"><h4 id="Fractal_geometry">Fractal geometry</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=21" title="Edit section: Fractal geometry">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mandelset_hires.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/220px-Mandelset_hires.png" decoding="async" width="220" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/330px-Mandelset_hires.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/440px-Mandelset_hires.png 2x" data-file-width="3121" data-file-height="2288" /></a><figcaption>The Mandelbrot set with the real and imaginary axes labeled.</figcaption></figure> The <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> is a popular example of a fractal formed on the complex plane. It is defined by plotting every location <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> where iterating the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}(z)=z^{2}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{c}(z)=z^{2}+c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191627a3eebdd6608c9b226786defc468b747502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:3.176ex;" alt="{\displaystyle f_{c}(z)=z^{2}+c}" /> does not <a href="/wiki/Diverge_(stability_theory)" class="mw-redirect" title="Diverge (stability theory)">diverge</a> when <a href="/wiki/Iteration" title="Iteration">iterated</a> infinitely. Similarly, <a href="/wiki/Julia_set" title="Julia set">Julia sets</a> have the same rules, except where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> remains constant. <div class="mw-heading mw-heading4"><h4 id="Triangles">Triangles</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=22" title="Edit section: Triangles">edit</a>]</div> Every triangle has a unique <a href="/wiki/Steiner_inellipse" title="Steiner inellipse">Steiner inellipse</a> – an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> inside the triangle and tangent to the midpoints of the three sides of the triangle. The <a href="/wiki/Focus_(geometry)" title="Focus (geometry)">foci</a> of a triangle's Steiner inellipse can be found as follows, according to <a href="/wiki/Marden%27s_theorem" title="Marden's theorem">Marden's theorem</a>:<a href="#cite_note-64">[56]</a><a href="#cite_note-65">[57]</a> Denote the triangle's vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Write the <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)(x-b)(x-c)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)(x-b)(x-c)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f8d9b149522168258a0bde389d0a53e9c6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.433ex; height:2.843ex;" alt="{\displaystyle (x-a)(x-b)(x-c)=0}" />, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse. <div class="mw-heading mw-heading3"><h3 id="Algebraic_number_theory">Algebraic number theory</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=23" title="Edit section: Algebraic number theory">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pentagon_construct.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/76/Pentagon_construct.gif" decoding="async" width="180" height="180" class="mw-file-element" data-file-width="180" data-file-height="180" /></a><figcaption>Construction of a regular pentagon <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">using straightedge and compass</a>.</figcaption></figure> As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. <a href="/wiki/Argumentum_a_fortiori" title="Argumentum a fortiori">A fortiori</a>, the same is true if the equation has rational coefficients. The roots of such equations are called <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a> – they are a principal object of study in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. Compared to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a8814b1ca454c488e409813988dd5dd906148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.923ex; height:3.343ex;" alt="{\displaystyle {\overline {\mathbb {Q} }}}" />, the algebraic closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />, which also contains all algebraic numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a> to the <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a> containing <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a>, it can be shown that it is not possible to construct a regular <a href="/wiki/Nonagon" title="Nonagon">nonagon</a> <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">using only compass and straightedge</a> – a purely geometric problem. Another example is the <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>; that is, numbers of the form x + iy, where x and y are integers, which can be used to classify <a href="/wiki/Fermat%27s_theorem_on_sums_of_two_squares" title="Fermat's theorem on sums of two squares">sums of squares</a>. <div class="mw-heading mw-heading3"><h3 id="Analytic_number_theory">Analytic number theory</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=24" title="Edit section: Analytic number theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></div> Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> ζ(s) is related to the distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. <div class="mw-heading mw-heading3"><h3 id="Improper_integrals">Improper integrals</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=25" title="Edit section: Improper integrals">edit</a>]</div> In applied fields, complex numbers are often used to compute certain real-valued <a href="/wiki/Improper_integral" title="Improper integral">improper integrals</a>, by means of complex-valued functions. Several methods exist to do this; see <a href="/wiki/Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">methods of contour integration</a>. <div class="mw-heading mw-heading3"><h3 id="Dynamic_equations">Dynamic equations</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=26" title="Edit section: Dynamic equations">edit</a>]</div> In <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, it is common to first find all complex roots r of the <a href="/wiki/Linear_differential_equation#Homogeneous_equation_with_constant_coefficients" title="Linear differential equation">characteristic equation</a> of a <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equation</a> or equation system and then attempt to solve the system in terms of base functions of the form f(t) = ert. Likewise, in <a href="/wiki/Difference_equations" class="mw-redirect" title="Difference equations">difference equations</a>, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form f(t) = rt. <div class="mw-heading mw-heading3"><h3 id="Linear_algebra">Linear algebra</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=27" title="Edit section: Linear algebra">edit</a>]</div> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is algebraically closed, any non-empty complex <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> has at least one (complex) <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a>. By comparison, real matrices do not always have real eigenvalues, for example <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrices</a> (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any real eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigendecomposition</a> is a useful tool for computing matrix powers and <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponentials</a>. Complex numbers often generalize concepts originally conceived in the real numbers. For example, the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> generalizes the <a href="/wiki/Transpose" title="Transpose">transpose</a>, <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian matrices</a> generalize <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>, and <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a> generalize <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a>. <div class="mw-heading mw-heading3"><h3 id="In_applied_mathematics">In applied mathematics</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=28" title="Edit section: In applied mathematics">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Control_theory">Control theory</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=29" title="Edit section: Control theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Complex_plane#Use_in_control_theory" title="Complex plane">Complex plane § Use in control theory</a></div> In <a href="/wiki/Control_theory" title="Control theory">control theory</a>, systems are often transformed from the <a href="/wiki/Time_domain" title="Time domain">time domain</a> to the complex <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> using the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>. The system's <a href="/wiki/Zeros_and_poles" title="Zeros and poles">zeros and poles</a> are then analyzed in the complex plane. The <a href="/wiki/Root_locus" class="mw-redirect" title="Root locus">root locus</a>, <a href="/wiki/Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a>, and <a href="/wiki/Nichols_plot" title="Nichols plot">Nichols plot</a> techniques all make use of the complex plane. In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are <ul><li>in the right half plane, it will be <a href="/wiki/Unstable" class="mw-redirect" title="Unstable">unstable</a>,</li> <li>all in the left half plane, it will be <a href="/wiki/BIBO_stability" title="BIBO stability">stable</a>,</li> <li>on the imaginary axis, it will have <a href="/wiki/Marginal_stability" title="Marginal stability">marginal stability</a>.</li></ul> If a system has zeros in the right half plane, it is a <a href="/wiki/Nonminimum_phase" class="mw-redirect" title="Nonminimum phase">nonminimum phase</a> system. <div class="mw-heading mw-heading4"><h4 id="Signal_analysis">Signal analysis</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=30" title="Edit section: Signal analysis">edit</a>]</div> Complex numbers are used in <a href="/wiki/Signal_analysis" class="mw-redirect" title="Signal analysis">signal analysis</a> and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a> of a given <a href="/wiki/Frequency" title="Frequency">frequency</a>, the absolute value |z| of the corresponding z is the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> and the <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> arg z is the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>. If <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\operatorname {Re} \{X(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=\operatorname {Re} \{X(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffdbcd895d1d9995bd3b58e3e84593fa2800d868" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.774ex; height:2.843ex;" alt="{\displaystyle x(t)=\operatorname {Re} \{X(t)\}}" /> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e065a79a4803b81d5dd1e938da8cfa8c8d8087" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.025ex; height:3.343ex;" alt="{\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}}" /> where ω represents the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> and the complex number A encodes the phase and amplitude as explained above. This use is also extended into <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a> and <a href="/wiki/Digital_image_processing" title="Digital image processing">digital image processing</a>, which use digital versions of Fourier analysis (and <a href="/wiki/Wavelet" title="Wavelet">wavelet</a> analysis) to transmit, <a href="/wiki/Data_compression" title="Data compression">compress</a>, restore, and otherwise process <a href="/wiki/Digital_data" title="Digital data">digital</a> <a href="/wiki/Sound" title="Sound">audio</a> signals, still images, and <a href="/wiki/Video" title="Video">video</a> signals. Another example, relevant to the two side bands of <a href="/wiki/Amplitude_modulation" title="Amplitude modulation">amplitude modulation</a> of AM radio, is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {Re} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {Re} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {Re} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {Re} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {Re} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {Re} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {Re} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {Re} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbed8f49057649de4c88600c3299463ff52b00e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.415ex; margin-bottom: -0.256ex; width:56.734ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {Re} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {Re} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {Re} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {Re} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}}" /> <div class="mw-heading mw-heading3"><h3 id="In_physics">In physics</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=31" title="Edit section: In physics">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Electromagnetism_and_electrical_engineering">Electromagnetism and electrical engineering</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=32" title="Edit section: Electromagnetism and electrical engineering">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></div> In <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> is used to analyze varying <a href="/wiki/Voltage" title="Voltage">voltages</a> and <a href="/wiki/Electric_current" title="Electric current">currents</a>. The treatment of <a href="/wiki/Resistor" title="Resistor">resistors</a>, <a href="/wiki/Capacitor" title="Capacitor">capacitors</a>, and <a href="/wiki/Inductor" title="Inductor">inductors</a> can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a>. This approach is called <a href="/wiki/Phasor" title="Phasor">phasor</a> calculus. In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use to denote <a href="/wiki/Electric_current" title="Electric current">electric current</a>, or, more particularly, i, which is generally in use to denote instantaneous electric current. Because the <a href="/wiki/Voltage" title="Voltage">voltage</a> in an AC <a href="/wiki/Electric_circuit" class="mw-redirect" title="Electric circuit">circuit</a> is oscillating, it can be represented as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db52b30a48d1206b576a033d782bf35752bb248f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.789ex; height:3.176ex;" alt="{\displaystyle V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),}" /> To obtain the measurable quantity, the real part is taken: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)=\operatorname {Re} (V)=\operatorname {Re} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=\operatorname {Re} (V)=\operatorname {Re} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9078e78decc9fdf5d57a237bbf756b9cc438a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.34ex; height:3.343ex;" alt="{\displaystyle v(t)=\operatorname {Re} (V)=\operatorname {Re} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.}" /> The complex-valued signal V(t) is called the <a href="/wiki/Analytic_signal" title="Analytic signal">analytic</a> representation of the real-valued, measurable signal v(t). <a href="#cite_note-66">[58]</a> <div class="mw-heading mw-heading4"><h4 id="Fluid_dynamics">Fluid dynamics</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=33" title="Edit section: Fluid dynamics">edit</a>]</div> In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, complex functions are used to describe <a href="/wiki/Potential_flow_in_two_dimensions" class="mw-redirect" title="Potential flow in two dimensions">potential flow in two dimensions</a>. <div class="mw-heading mw-heading4"><h4 id="Quantum_mechanics">Quantum mechanics</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=34" title="Edit section: Quantum mechanics">edit</a>]</div> The complex number field is intrinsic to the <a href="/wiki/Mathematical_formulations_of_quantum_mechanics" class="mw-redirect" title="Mathematical formulations of quantum mechanics">mathematical formulations of quantum mechanics</a>, where complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> and Heisenberg's <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> – make use of complex numbers. <div class="mw-heading mw-heading4"><h4 id="Relativity">Relativity</h4>[<a href="/w/index.php?title=Complex_number&action=edit&section=35" title="Edit section: Relativity">edit</a>]</div> In <a href="/wiki/Special_relativity" title="Special relativity">special</a> and <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, some formulas for the metric on <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is <a href="/wiki/Wick_rotation" title="Wick rotation">used in an essential way</a> in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.) Complex numbers are essential to <a href="/wiki/Spinor" title="Spinor">spinors</a>, which are a generalization of the <a href="/wiki/Tensor" title="Tensor">tensors</a> used in relativity. <div class="mw-heading mw-heading2"><h2 id="Characterizations,_generalizations_and_related_notions">Characterizations, generalizations and related notions</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=36" title="Edit section: Characterizations, generalizations and related notions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Algebraic_characterization">Algebraic characterization</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=37" title="Edit section: Algebraic characterization">edit</a>]</div> The field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> has the following three properties: <ul><li>First, it has <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one).</li> <li>Second, its <a href="/wiki/Transcendence_degree" class="mw-redirect" title="Transcendence degree">transcendence degree</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />, the <a href="/wiki/Prime_field" class="mw-redirect" title="Prime field">prime field</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" /> is the <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>.</li> <li>Third, it is <a href="/wiki/Algebraically_closed" class="mw-redirect" title="Algebraically closed">algebraically closed</a> (see above).</li></ul> It can be shown that any field having these properties is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> (as a field) to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /> For example, the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" /> of the <a href="/wiki/P-adic_number" title="P-adic number">p-adic number</a> also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).<a href="#cite_note-67">[59]</a> Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is isomorphic to the field of complex <a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a>. However, specifying an isomorphism requires the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. Another consequence of this algebraic characterization is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> contains many proper subfields that are isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />. <div class="mw-heading mw-heading3"><h3 id="Characterization_as_a_topological_field">Characterization as a topological field</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=38" title="Edit section: Characterization as a topological field">edit</a>]</div> The preceding characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> describes only the algebraic aspects of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /> That is to say, the properties of <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">nearness</a> and <a href="/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">continuity</a>, which matter in areas such as <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> and <a href="/wiki/Topology" title="Topology">topology</a>, are not dealt with. The following description of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> as a <a href="/wiki/Topological_ring" title="Topological ring">topological field</a> (that is, a field that is equipped with a <a href="/wiki/Topological_space" title="Topological space">topology</a>, which allows the notion of convergence) does take into account the topological properties. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions: <ul><li>P is closed under addition, multiplication and taking inverses.</li> <li>If x and y are distinct elements of P, then either x − y or y − x is in P.</li> <li>If S is any nonempty subset of P, then S + P = x + P for some x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /></li></ul> Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> has a nontrivial <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutive</a> <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> x ↦ x* (namely the complex conjugation), such that x x* is in P for any nonzero x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /> Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (y − x)(y − x)* ∈ P } as a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a>, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /> The only <a href="/wiki/Connected_space" title="Connected space">connected</a> <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="/wiki/Topological_ring" title="Topological ring">topological fields</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /> This gives another characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> as a topological field, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> can be distinguished from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> because the nonzero complex numbers are <a href="/wiki/Connected_space" title="Connected space">connected</a>, while the nonzero real numbers are not.<a href="#cite_note-FOOTNOTEBourbaki1998§VIII.4-68">[60]</a> <div class="mw-heading mw-heading3"><h3 id="Other_number_systems">Other number systems</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=39" title="Edit section: Other number systems">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a>, <a href="/wiki/Quaternion" title="Quaternion">Quaternion</a>, and <a href="/wiki/Octonion" title="Octonion">Octonion</a></div> <table class="wikitable"> <caption>Number systems </caption> <tbody><tr> <th> </th> <th>rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> </th> <th>real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> </th> <th>complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> </th> <th>quaternions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" /> </th> <th>octonions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }" /> </th> <th>sedenions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" /> </th></tr> <tr> <th><a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th><a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> as an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />-vector space </th> <td>[does not apply]</td> <td>1</td> <td>2</td> <td>4</td> <td>8</td> <td>16 </td></tr> <tr> <th><a href="/wiki/Ordered_field" title="Ordered field">ordered</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th>multiplication commutative (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=yx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b203fa309e89fccdbba22909c8418f6b229779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.069ex; height:2.009ex;" alt="{\displaystyle xy=yx}" />) </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th>multiplication associative (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xy)z=x(yz)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)z=x(yz)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb185761d3d71a9a59cf8ed17b9a40c518e08ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.864ex; height:2.843ex;" alt="{\displaystyle (xy)z=x(yz)}" />) </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr> <tr> <th><a href="/wiki/Normed_division_algebra" class="mw-redirect" title="Normed division algebra">normed division algebra</a> (over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />) </th> <td>[does not apply]</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td></tr></tbody></table> The process of extending the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> of reals to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is an instance of the Cayley–Dickson construction. Applying this construction iteratively to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> then yields the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, the <a href="/wiki/Octonion" title="Octonion">octonions</a>,<a href="#cite_note-69">[61]</a> the <a href="/wiki/Sedenion" title="Sedenion">sedenions</a>, and the <a href="/wiki/Trigintaduonion" title="Trigintaduonion">trigintaduonions</a>. This construction turns out to diminish the structural properties of the involved number systems. Unlike the reals, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is not an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a>, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i2 = −1 precludes the existence of an <a href="/wiki/Total_order" title="Total order">ordering</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" /><a href="#cite_note-FOOTNOTEApostol198125-70">[62]</a> Passing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> to the quaternions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" /> loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all <a href="/wiki/Normed_division_algebra" class="mw-redirect" title="Normed division algebra">normed division algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />. By <a href="/wiki/Hurwitz%27s_theorem_(normed_division_algebras)" class="mw-redirect" title="Hurwitz's theorem (normed division algebras)">Hurwitz's theorem</a> they are the only ones; the <a href="/wiki/Sedenion" title="Sedenion">sedenions</a>, the next step in the Cayley–Dickson construction, fail to have this structure. The Cayley–Dickson construction is closely related to the <a href="/wiki/Regular_representation" title="Regular representation">regular representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" /> thought of as an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />-<a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">algebra</a> (an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />-vector space with a multiplication), with respect to the basis (1, i). This means the following: the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />-linear map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>w</mi> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271354b63f808b0b493fc7da9fb0bbe791c3dea4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.796ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}}" /> for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45552f82e2336286287937c9fd47a92fec363f36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.835ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},}" /> that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> in the 2 × 2 real matrices, it is not the only one. Any matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mo>−</mo> <mi>p</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d829f1d6ebf86155a275bfb2dc65d67b62b886b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.015ex; height:6.176ex;" alt="{\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0}" /> has the property that its square is the negative of the identity matrix: J2 = −I. Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z=aI+bJ:a,b\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mi>I</mi> <mo>+</mo> <mi>b</mi> <mi>J</mi> <mo>:</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{z=aI+bJ:a,b\in \mathbb {R} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a5d870b8bd7b1820d6da1b8686eab4abbe5bd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.94ex; height:2.843ex;" alt="{\displaystyle \{z=aI+bJ:a,b\in \mathbb {R} \}}" /> is also isomorphic to the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" /> and gives an alternative complex structure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}" /> This is generalized by the notion of a <a href="/wiki/Linear_complex_structure" title="Linear complex structure">linear complex structure</a>. <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a> also generalize <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" /> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" /> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d178e5ac94e706fdb8d8733d567b7c087b23195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {H} ,}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdb835d3672e3531f7356ff7327bc996ec44aa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} .}" /> For example, this notion contains the <a href="/wiki/Split-complex_number" title="Split-complex number">split-complex numbers</a>, which are elements of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x]/(x^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [x]/(x^{2}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29edbdd7a09968cb2fd42397bcab00406e77854c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.66ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} [x]/(x^{2}-1)}" /> (as opposed to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x]/(x^{2}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [x]/(x^{2}+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0ade67281f83ef6b6b7f43bf783c081adb1fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.66ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} [x]/(x^{2}+1)}" /> for complex numbers). In this ring, the equation a2 = 1 has four solutions. The field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is the completion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}" /> the field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, with respect to the usual <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>. Other choices of <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metrics</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> lead to the fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" /> of <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> (for any <a href="/wiki/Prime_number" title="Prime number">prime number</a> p), which are thereby analogous to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />. There are no other nontrivial ways of completing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d194e3e8fce9335ed524db967666b4f02fb523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p},}" /> by <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a>. The algebraic closures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} _{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} _{p}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7067dfc2452aaa42321439c9e7aed4641686f4c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.982ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathbb {Q} _{p}}}}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" /> still carry a norm, but (unlike <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />) are not complete with respect to it. The completion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} _{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} _{p}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7067dfc2452aaa42321439c9e7aed4641686f4c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.982ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathbb {Q} _{p}}}}" /> turns out to be algebraically closed. By analogy, the field is called p-adic complex numbers. The fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" /> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d194e3e8fce9335ed524db967666b4f02fb523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p},}" /> and their finite field extensions, including <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" /> are called <a href="/wiki/Local_field" title="Local field">local fields</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=40" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Complex_numbers" class="extiw" title="commons:Category:Complex numbers">Complex numbers</a>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <a href="https://en.wikiversity.org/wiki/Complex_Numbers" class="extiw" title="v:Complex Numbers">Complex Numbers</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Calculus/Complex_numbers" class="extiw" title="wikibooks:Calculus/Complex numbers">Calculus/Complex numbers</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikisource-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/40px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/60px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/120px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 Encyclopædia Britannica</a> article "<a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Number/Complex_Numbers" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Number/Complex Numbers">Number/Complex Numbers</a>".</div></div> </div> <ul><li><a href="/wiki/Analytic_continuation" title="Analytic continuation">Analytic continuation</a></li> <li><a href="/wiki/Circular_motion#Using_complex_numbers" title="Circular motion">Circular motion using complex numbers</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex-base system</a></li> <li><a href="/wiki/Complex_coordinate_space" title="Complex coordinate space">Complex coordinate space</a></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex geometry</a></li> <li><a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">Geometry of numbers</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex number</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a href="/wiki/Geometric_algebra#Unit_pseudoscalars" title="Geometric algebra">Geometric algebra</a> (which includes the complex plane as the 2-dimensional <a href="/wiki/Spinor#Two_dimensions" title="Spinor">spinor</a> subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{2}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{2}^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6023850da07089febe34ebd02728b8c7a3e05cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.906ex; height:3.009ex;" alt="{\displaystyle {\mathcal {G}}_{2}^{+}}" />)</li> <li><a href="/wiki/Unit_complex_number" class="mw-redirect" title="Unit complex number">Unit complex number</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=41" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFSolomentsev2001">Solomentsev 2001</a>: "The plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" /> whose points are identified with the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> is called the complex plane ... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel". </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.<a href="#cite_note-27">[26]</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> It has been proved that imaginary numbers necessarily appear in the cubic formula when the equation has three real, different roots by Pierre Laurent Wantzel in 1843, Vincenzo Mollame in 1890, Otto Hölder in 1891, and Adolf Kneser in 1892. Paolo Ruffini also provided an incomplete proof in 1799.——S. Confalonieri (2015)<a href="#cite_note-Casus-30">[28]</a> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a href="#CITEREFArgand1814">Argand 1814</a>, p. 204 defines the modulus of a complex number but he doesn't name it: "Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=m+n{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=m+n{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7094e9db544b53538975f5459e82cd1b8ebd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.51ex; height:3.009ex;" alt="{\displaystyle a=m+n{\sqrt {-1}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> étant réels, on devra entendre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{'}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc5961b7c2f7efd7f3b1077f7bcc537e64f43cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.994ex; height:2.009ex;" alt="{\displaystyle a_{'}}" /> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b94eab63e63bb9ba12ea5f72788829ce5320b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.721ex; height:3.509ex;" alt="{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}" />." [In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=m+n{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=m+n{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7094e9db544b53538975f5459e82cd1b8ebd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.51ex; height:3.009ex;" alt="{\displaystyle a=m+n{\sqrt {-1}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> being real, one should understand that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{'}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc5961b7c2f7efd7f3b1077f7bcc537e64f43cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.994ex; height:2.009ex;" alt="{\displaystyle a_{'}}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b94eab63e63bb9ba12ea5f72788829ce5320b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.721ex; height:3.509ex;" alt="{\displaystyle a'={\sqrt {m^{2}+n^{2}}}}" />.] <a href="#CITEREFArgand1814">Argand 1814</a>, p. 208 defines and names the module and the direction factor of a complex number: "... <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {m^{2}+n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {m^{2}+n^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b0a9c9443d7b6a91034a67fafd8a1fefe1d156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.036ex; height:3.509ex;" alt="{\displaystyle a={\sqrt {m^{2}+n^{2}}}}" /> pourrait être appelé le module de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd81d057bc6b40c69ed7dd94e920562c63eafe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-1}}}" />, et représenterait la grandeur absolue de la ligne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd81d057bc6b40c69ed7dd94e920562c63eafe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-1}}}" />, tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction." [... <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {m^{2}+n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {m^{2}+n^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b0a9c9443d7b6a91034a67fafd8a1fefe1d156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.036ex; height:3.509ex;" alt="{\displaystyle a={\sqrt {m^{2}+n^{2}}}}" /> could be called the module of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd81d057bc6b40c69ed7dd94e920562c63eafe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-1}}}" /> and would represent the absolute size of the line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-1}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-1}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376768abf04feb3e23dbb75d9430310038fe4c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.008ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-1}}\,,}" /> (Argand represented complex numbers as vectors.) whereas the other factor [namely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{\sqrt {a^{2}+b^{2}}}}+{\tfrac {b}{\sqrt {a^{2}+b^{2}}}}{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{\sqrt {a^{2}+b^{2}}}}+{\tfrac {b}{\sqrt {a^{2}+b^{2}}}}{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd288ad4265a17ac15fd78142b169651cbf17cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.739ex; height:5.676ex;" alt="{\displaystyle {\tfrac {a}{\sqrt {a^{2}+b^{2}}}}+{\tfrac {b}{\sqrt {a^{2}+b^{2}}}}{\sqrt {-1}}}" />], whose module is unity [1], would represent its direction.] </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Gauss writes:<a href="#cite_note-54">[50]</a> "Quemadmodum scilicet arithmetica sublimior in quaestionibus hactenus pertractatis inter solos numeros integros reales versatur, ita theoremata circa residua biquadratica tunc tantum in summa simplicitate ac genuina venustate resplendent, quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i, pro more quantitatem imaginariam <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" />, atque a, b indefinite omnes numeros reales integros inter -<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /> et +<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" />." [Of course just as the higher arithmetic has been investigated so far in problems only among real integer numbers, so theorems regarding biquadratic residues then shine in greatest simplicity and genuine beauty, when the field of arithmetic is extended to imaginary quantities, so that, without restrictions on it, numbers of the form a + bi — i denoting by convention the imaginary quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" />, and the variables a, b [denoting] all real integer numbers between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" /> — constitute an object.] </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> Gauss:<a href="#cite_note-56">[51]</a> "Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeantur." [We will call such numbers [namely, numbers of the form a + bi ] "complex integer numbers", so that real [numbers] are regarded not as the opposite of complex [numbers] but [as] a type [of number that] is, so to speak, contained within them.] </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Gauss:<a href="#cite_note-58">[52]</a> "Productum numeri complexi per numerum ipsi conjunctum utriusque normam vocamus. Pro norma itaque numeri realis, ipsius quadratum habendum est." [We call a "norm" the product of a complex number [for example, a + ib ] with its conjugate [a - ib ]. Therefore the square of a real number should be regarded as its norm.] </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other <a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a> thru the origin. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Complex_number&action=edit&section=42" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see <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="CITEREFBourbaki1998" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1998). "Foundations of Mathematics § Logic: Set theory". Elements of the History of Mathematics. Springer. pp. 18–24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foundations+of+Mathematics+%C2%A7+Logic%3A+Set+theory&rft.btitle=Elements+of+the+History+of+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E18-%3C%2Fspan%3E24&rft.pub=Springer&rft.date=1998&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> "Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.", <a href="#CITEREFPenrose2005">Penrose 2005</a>, pp.72–73. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAxler2010" class="citation book cs1">Axler, Sheldon (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/collegealgebrawi00axle">College algebra</a>. Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/collegealgebrawi00axle/page/n285">262</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470470770" title="Special:BookSources/9780470470770"><bdi>9780470470770</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=262&rft.pub=Wiley&rft.date=2010&rft.isbn=9780470470770&rft.aulast=Axler&rft.aufirst=Sheldon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcollegealgebrawi00axle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSpiegelLipschutzSchillerSpellman2009" class="citation book cs1">Spiegel, M.R.; Lipschutz, S.; Schiller, J.J.; Spellman, D. (14 April 2009). Complex Variables. Schaum's Outline Series (2nd ed.). McGraw Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-161569-3" title="Special:BookSources/978-0-07-161569-3"><bdi>978-0-07-161569-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Variables&rft.series=Schaum%27s+Outline+Series&rft.edition=2nd&rft.pub=McGraw+Hill&rft.date=2009-04-14&rft.isbn=978-0-07-161569-3&rft.aulast=Spiegel&rft.aufirst=M.R.&rft.au=Lipschutz%2C+S.&rft.au=Schiller%2C+J.J.&rft.au=Spellman%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFAufmannBarkerNation2007">Aufmann, Barker & Nation 2007</a>, p. 66, Chapter P </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPedoe1988" class="citation book cs1"><a href="/wiki/Daniel_Pedoe" title="Daniel Pedoe">Pedoe, Dan</a> (1988). Geometry: A comprehensive course. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65812-4" title="Special:BookSources/978-0-486-65812-4"><bdi>978-0-486-65812-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+A+comprehensive+course&rft.pub=Dover&rft.date=1988&rft.isbn=978-0-486-65812-4&rft.aulast=Pedoe&rft.aufirst=Dan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-:2-7">^ <a href="#cite_ref-:2_7-0">a</a> <a href="#cite_ref-:2_7-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ComplexNumber.html">"Complex Number"</a>. mathworld.wolfram.com. Retrieved 12 August 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Complex+Number&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FComplexNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-Campbell_1911-9"><a href="#cite_ref-Campbell_1911_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCampbell1911" class="citation journal cs1"><a href="/wiki/George_Ashley_Campbell" title="George Ashley Campbell">Campbell, George Ashley</a> (April 1911). <a rel="nofollow" class="external text" href="https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf">"Cisoidal oscillations"</a> (PDF). <a href="/wiki/Proceedings_of_the_American_Institute_of_Electrical_Engineers" class="mw-redirect" title="Proceedings of the American Institute of Electrical Engineers">Proceedings of the American Institute of Electrical Engineers</a>. XXX (1–6). <a href="/wiki/American_Institute_of_Electrical_Engineers" title="American Institute of Electrical Engineers">American Institute of Electrical Engineers</a>: 789–824 [Fig. 13 on p. 810]. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPAIEE.1911.6659711">10.1109/PAIEE.1911.6659711</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51647814">51647814</a>. Retrieved 24 June 2023. p. 789: <q>The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Institute+of+Electrical+Engineers&rft.atitle=Cisoidal+oscillations&rft.volume=XXX&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E6&rft.pages=789-824+Fig.+13+on+p.+810&rft.date=1911-04&rft_id=info%3Adoi%2F10.1109%2FPAIEE.1911.6659711&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51647814%23id-name%3DS2CID&rft.aulast=Campbell&rft.aufirst=George+Ashley&rft_id=https%3A%2F%2Fia800708.us.archive.org%2Fview_archive.php%3Farchive%3D%2F28%2Fitems%2Fcrossref-pre-1923-scholarly-works%2F10.1109%25252Fpaiee.1910.6660428.zip%26file%3D10.1109%25252Fpaiee.1911.6659711.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-Brown-Churchill_1996-10"><a href="#cite_ref-Brown-Churchill_1996_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrownChurchill1996" class="citation book cs1">Brown, James Ward; Churchill, Ruel V. (1996). Complex variables and applications (6 ed.). New York, USA: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-912147-9" title="Special:BookSources/978-0-07-912147-9"><bdi>978-0-07-912147-9</bdi></a>. p. 2: <q>In electrical engineering, the letter j is used instead of i.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+variables+and+applications&rft.place=New+York%2C+USA&rft.pages=2&rft.edition=6&rft.pub=McGraw-Hill&rft.date=1996&rft.isbn=978-0-07-912147-9&rft.aulast=Brown&rft.aufirst=James+Ward&rft.au=Churchill%2C+Ruel+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEApostol198115–16-11"><a href="#cite_ref-FOOTNOTEApostol198115–16_11-0">^</a> <a href="#CITEREFApostol1981">Apostol 1981</a>, pp. 15–16. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFApostol1981">Apostol 1981</a>, pp. 15–16 </li> <li id="cite_note-FOOTNOTEApostol198118-13"><a href="#cite_ref-FOOTNOTEApostol198118_13-0">^</a> <a href="#CITEREFApostol1981">Apostol 1981</a>, p. 18. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilliam_Ford2014" class="citation book cs1">William Ford (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OODs2mkOOqAC">Numerical Linear Algebra with Applications: Using MATLAB and Octave</a> (reprinted ed.). Academic Press. p. 570. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-394784-0" title="Special:BookSources/978-0-12-394784-0"><bdi>978-0-12-394784-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Linear+Algebra+with+Applications%3A+Using+MATLAB+and+Octave&rft.pages=570&rft.edition=reprinted&rft.pub=Academic+Press&rft.date=2014&rft.isbn=978-0-12-394784-0&rft.au=William+Ford&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOODs2mkOOqAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OODs2mkOOqAC&pg=PA570">Extract of page 570</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDennis_ZillJacqueline_Dewar2011" class="citation book cs1">Dennis Zill; Jacqueline Dewar (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TLgjLBeY55YC">Precalculus with Calculus Previews: Expanded Volume</a> (revised ed.). Jones & Bartlett Learning. p. 37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-6631-3" title="Special:BookSources/978-0-7637-6631-3"><bdi>978-0-7637-6631-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus+with+Calculus+Previews%3A+Expanded+Volume&rft.pages=37&rft.edition=revised&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2011&rft.isbn=978-0-7637-6631-3&rft.au=Dennis+Zill&rft.au=Jacqueline+Dewar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTLgjLBeY55YC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TLgjLBeY55YC&pg=PA37">Extract of page 37</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Other authors, including <a href="#CITEREFEbbinghausHermesHirzebruchKoecher1991">Ebbinghaus et al. 1991</a>, §6.1, chose the argument to be in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}" />. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKasana2005" class="citation book cs1">Kasana, H.S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14">"Chapter 1"</a>. Complex Variables: Theory And Applications (2nd ed.). PHI Learning Pvt. Ltd. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-2641-5" title="Special:BookSources/978-81-203-2641-5"><bdi>978-81-203-2641-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1&rft.btitle=Complex+Variables%3A+Theory+And+Applications&rft.pages=14&rft.edition=2nd&rft.pub=PHI+Learning+Pvt.+Ltd&rft.date=2005&rft.isbn=978-81-203-2641-5&rft.aulast=Kasana&rft.aufirst=H.S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrFhiJqkrALIC%26pg%3DPA14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNilssonRiedel2008" class="citation book cs1">Nilsson, James William; Riedel, Susan A. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338">"Chapter 9"</a>. Electric circuits (8th ed.). Prentice Hall. p. 338. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-198925-2" title="Special:BookSources/978-0-13-198925-2"><bdi>978-0-13-198925-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+9&rft.btitle=Electric+circuits&rft.pages=338&rft.edition=8th&rft.pub=Prentice+Hall&rft.date=2008&rft.isbn=978-0-13-198925-2&rft.aulast=Nilsson&rft.aufirst=James+William&rft.au=Riedel%2C+Susan+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsxmM8RFL99wC%26pg%3DPA338&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLloyd_James_Peter_Kilford2015" class="citation book cs1">Lloyd James Peter Kilford (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qDk8DQAAQBAJ">Modular Forms: A Classical And Computational Introduction</a> (2nd ed.). World Scientific Publishing Company. p. 112. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-78326-547-3" title="Special:BookSources/978-1-78326-547-3"><bdi>978-1-78326-547-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modular+Forms%3A+A+Classical+And+Computational+Introduction&rft.pages=112&rft.edition=2nd&rft.pub=World+Scientific+Publishing+Company&rft.date=2015&rft.isbn=978-1-78326-547-3&rft.au=Lloyd+James+Peter+Kilford&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqDk8DQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qDk8DQAAQBAJ&pg=PA112">Extract of page 112</a> </li> <li id="cite_note-Bourbaki_1998_loc=§VIII.1-20"><a href="#cite_ref-Bourbaki_1998_loc=§VIII.1_20-0">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §VIII.1 </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKline" class="citation book cs1">Kline, Morris. A history of mathematical thought, volume 1. p. 253.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+mathematical+thought%2C+volume+1&rft.pages=253&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJurij." class="citation book cs1">Jurij., Kovič. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1080410598">Tristan Needham, Visual Complex Analysis, Oxford University Press Inc., New York, 1998, 592 strani</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1080410598">1080410598</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tristan+Needham%2C+Visual+Complex+Analysis%2C+Oxford+University+Press+Inc.%2C+New+York%2C+1998%2C+592+strani&rft_id=info%3Aoclcnum%2F1080410598&rft.aulast=Jurij.&rft.aufirst=Kovi%C4%8D&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F1080410598&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> O'Connor and Robertson (2016), "Girolamo Cardano." </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998. </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKatz2004" class="citation book cs1">Katz, Victor J. (2004). "9.1.4". A History of Mathematics, Brief Version. <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-16193-2" title="Special:BookSources/978-0-321-16193-2"><bdi>978-0-321-16193-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=9.1.4&rft.btitle=A+History+of+Mathematics%2C+Brief+Version&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-321-16193-2&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHamilton1844" class="citation journal cs1">Hamilton, Wm. (1844). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=njp.32101040410779&view=1up&seq=454">"On a new species of imaginary quantities connected with a theory of quaternions"</a>. Proceedings of the Royal Irish Academy. 2: 424–434.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Irish+Academy&rft.atitle=On+a+new+species+of+imaginary+quantities+connected+with+a+theory+of+quaternions&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E424-%3C%2Fspan%3E434&rft.date=1844&rft.aulast=Hamilton&rft.aufirst=Wm.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dnjp.32101040410779%26view%3D1up%26seq%3D454&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCynthia_Y._Young2017" class="citation book cs1">Cynthia Y. Young (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ">Trigonometry</a> (4th ed.). John Wiley & Sons. p. 406. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-44520-3" title="Special:BookSources/978-1-119-44520-3"><bdi>978-1-119-44520-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometry&rft.pages=406&rft.edition=4th&rft.pub=John+Wiley+%26+Sons&rft.date=2017&rft.isbn=978-1-119-44520-3&rft.au=Cynthia+Y.+Young&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D476ZDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA406">Extract of page 406</a> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNahin2007" class="citation book cs1">Nahin, Paul J. (2007). <a rel="nofollow" class="external text" href="http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284">An Imaginary Tale: The Story of √−1</a>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12798-9" title="Special:BookSources/978-0-691-12798-9"><bdi>978-0-691-12798-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284">Archived</a> from the original on 12 October 2012. Retrieved 20 April 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Imaginary+Tale%3A+The+Story+of+%E2%88%9A%E2%88%921&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-12798-9&rft.aulast=Nahin&rft.aufirst=Paul+J.&rft_id=http%3A%2F%2Fmathforum.org%2Fkb%2Fthread.jspa%3FforumID%3D149%26threadID%3D383188%26messageID%3D1181284&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-Casus-30">^ <a href="#cite_ref-Casus_30-0">a</a> <a href="#cite_ref-Casus_30-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConfalonieri2015" class="citation book cs1">Confalonieri, Sara (2015). The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations: Gerolamo Cardano's De Regula Aliza. Springer. pp. 15–16 (note 26). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3658092757" title="Special:BookSources/978-3658092757"><bdi>978-3658092757</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+Unattainable+Attempt+to+Avoid+the+Casus+Irreducibilis+for+Cubic+Equations%3A+Gerolamo+Cardano%27s+De+Regula+Aliza&rft.pages=15-16+%28note+26%29&rft.pub=Springer&rft.date=2015&rft.isbn=978-3658092757&rft.aulast=Confalonieri&rft.aufirst=Sara&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDescartes1954" class="citation book cs1"><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a> (1954) [1637]. <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofrenede00rend">La Géométrie | The Geometry of René Descartes with a facsimile of the first edition</a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60068-0" title="Special:BookSources/978-0-486-60068-0"><bdi>978-0-486-60068-0</bdi></a>. Retrieved 20 April 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=La+G%C3%A9om%C3%A9trie+%26%23124%3B+The+Geometry+of+Ren%C3%A9+Descartes+with+a+facsimile+of+the+first+edition&rft.pub=Dover+Publications&rft.date=1954&rft.isbn=978-0-486-60068-0&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofrenede00rend&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph_Mazur2016" class="citation book cs1">Joseph Mazur (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O3CYDwAAQBAJ">Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers</a> (reprinted ed.). Princeton University Press. p. 138. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-17337-5" title="Special:BookSources/978-0-691-17337-5"><bdi>978-0-691-17337-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enlightening+Symbols%3A+A+Short+History+of+Mathematical+Notation+and+Its+Hidden+Powers&rft.pages=138&rft.edition=reprinted&rft.pub=Princeton+University+Press&rft.date=2016&rft.isbn=978-0-691-17337-5&rft.au=Joseph+Mazur&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DO3CYDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O3CYDwAAQBAJ&pg=PA138">Extract of page 138</a> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBryan_Bunch2012" class="citation book cs1">Bryan Bunch (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jUTCAgAAQBAJ">Mathematical Fallacies and Paradoxes</a> (reprinted, revised ed.). Courier Corporation. p. 32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13793-3" title="Special:BookSources/978-0-486-13793-3"><bdi>978-0-486-13793-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Fallacies+and+Paradoxes&rft.pages=32&rft.edition=reprinted%2C+revised&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-13793-3&rft.au=Bryan+Bunch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjUTCAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA32">Extract of page 32</a> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuler1748" class="citation book cs1 cs1-prop-foreign-lang-source">Euler, Leonard (1748). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jQ1bAAAAQAAJ&pg=PA104">Introductio in Analysin Infinitorum</a> [Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductio+in+Analysin+Infinitorum&rft.place=Lucerne%2C+Switzerland&rft.pages=104&rft.pub=Marc+Michel+Bosquet+%26+Co.&rft.date=1748&rft.aulast=Euler&rft.aufirst=Leonard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjQ1bAAAAQAAJ%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWessel1799" class="citation journal cs1 cs1-prop-foreign-lang-source">Wessel, Caspar (1799). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=ien.35556000979690&view=1up&seq=561">"Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning"</a> [On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons]. Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] (in Danish). 5: 469–518.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nye+Samling+af+det+Kongelige+Danske+Videnskabernes+Selskabs+Skrifter+%5BNew+Collection+of+the+Writings+of+the+Royal+Danish+Science+Society%5D&rft.atitle=Om+Directionens+analytiske+Betegning%2C+et+Forsog%2C+anvendt+fornemmelig+til+plane+og+sph%C3%A6riske+Polygoners+Oplosning&rft.volume=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E469-%3C%2Fspan%3E518&rft.date=1799&rft.aulast=Wessel&rft.aufirst=Caspar&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dien.35556000979690%26view%3D1up%26seq%3D561&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWallis1685" class="citation book cs1">Wallis, John (1685). <a rel="nofollow" class="external text" href="https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/H3GRV5AU/pageimg&start=291&mode=imagepath&pn=291">A Treatise of Algebra, Both Historical and Practical ...</a> London, England: printed by John Playford, for Richard Davis. pp. 264–273.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+of+Algebra%2C+Both+Historical+and+Practical+...&rft.place=London%2C+England&rft.pages=%3Cspan+class%3D%22nowrap%22%3E264-%3C%2Fspan%3E273&rft.pub=printed+by+John+Playford%2C+for+Richard+Davis&rft.date=1685&rft.aulast=Wallis&rft.aufirst=John&rft_id=https%3A%2F%2Fecho.mpiwg-berlin.mpg.de%2FECHOdocuView%3Furl%3D%2Fpermanent%2Flibrary%2FH3GRV5AU%2Fpageimg%26start%3D291%26mode%3Dimagepath%26pn%3D291&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArgand1806" class="citation book cs1 cs1-prop-foreign-lang-source">Argand (1806). <a rel="nofollow" class="external text" href="http://www.bibnum.education.fr/mathematiques/geometrie/essai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons">Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques</a> [Essay on a way to represent complex quantities by geometric constructions] (in French). Paris, France: Madame Veuve Blanc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essai+sur+une+mani%C3%A8re+de+repr%C3%A9senter+les+quantit%C3%A9s+imaginaires+dans+les+constructions+g%C3%A9om%C3%A9triques&rft.place=Paris%2C+France&rft.pub=Madame+Veuve+Blanc&rft.date=1806&rft.au=Argand&rft_id=http%3A%2F%2Fwww.bibnum.education.fr%2Fmathematiques%2Fgeometrie%2Fessai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> Gauss, Carl Friedrich (1799) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g3VaAAAAcAAJ&pg=PP1">"Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse."</a> [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin) </li> <li id="cite_note-Ewald-40"><a href="#cite_ref-Ewald_40-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEwald1996" class="citation book cs1">Ewald, William B. (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA313">From Kant to Hilbert: A Source Book in the Foundations of Mathematics</a>. Vol. 1. Oxford University Press. p. 313. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198505358" title="Special:BookSources/9780198505358"><bdi>9780198505358</bdi></a>. Retrieved 18 March 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Kant+to+Hilbert%3A+A+Source+Book+in+the+Foundations+of+Mathematics&rft.pages=313&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=9780198505358&rft.aulast=Ewald&rft.aufirst=William+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrykSDAAAQBAJ%26pg%3DPA313&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEGauss1831-41"><a href="#cite_ref-FOOTNOTEGauss1831_41-0">^</a> <a href="#CITEREFGauss1831">Gauss 1831</a>. </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Buee/">"Adrien Quentin Buée (1745–1845): MacTutor"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Adrien+Quentin+Bu%C3%A9e+%281745%E2%80%931845%29%3A+MacTutor&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FBuee%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBuée1806" class="citation journal cs1 cs1-prop-foreign-lang-source">Buée (1806). <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1806.0003">"Mémoire sur les quantités imaginaires"</a> [Memoir on imaginary quantities]. Philosophical Transactions of the Royal Society of London (in French). 96: 23–88. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1806.0003">10.1098/rstl.1806.0003</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:110394048">110394048</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=M%C3%A9moire+sur+les+quantit%C3%A9s+imaginaires&rft.volume=96&rft.pages=%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E88&rft.date=1806&rft_id=info%3Adoi%2F10.1098%2Frstl.1806.0003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A110394048%23id-name%3DS2CID&rft.au=Bu%C3%A9e&rft_id=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2Fpdf%2F10.1098%2Frstl.1806.0003&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMourey1861" class="citation book cs1 cs1-prop-foreign-lang-source">Mourey, C.V. (1861). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_8YxKAAAAYAAJ">La vraies théore des quantités négatives et des quantités prétendues imaginaires</a> [The true theory of negative quantities and of alleged imaginary quantities] (in French). Paris, France: Mallet-Bachelier.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=La+vraies+th%C3%A9ore+des+quantit%C3%A9s+n%C3%A9gatives+et+des+quantit%C3%A9s+pr%C3%A9tendues+imaginaires&rft.place=Paris%2C+France&rft.pub=Mallet-Bachelier&rft.date=1861&rft.aulast=Mourey&rft.aufirst=C.V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_8YxKAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> 1861 reprint of 1828 original. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarren1828" class="citation book cs1">Warren, John (1828). <a rel="nofollow" class="external text" href="https://archive.org/details/treatiseongeomet00warrrich">A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities</a>. Cambridge, England: Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+the+Geometrical+Representation+of+the+Square+Roots+of+Negative+Quantities&rft.place=Cambridge%2C+England&rft.pub=Cambridge+University+Press&rft.date=1828&rft.aulast=Warren&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftreatiseongeomet00warrrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarren1829" class="citation journal cs1">Warren, John (1829). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1829.0022">"Consideration of the objections raised against the geometrical representation of the square roots of negative quantities"</a>. Philosophical Transactions of the Royal Society of London. 119: 241–254. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1829.0022">10.1098/rstl.1829.0022</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186211638">186211638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=Consideration+of+the+objections+raised+against+the+geometrical+representation+of+the+square+roots+of+negative+quantities&rft.volume=119&rft.pages=%3Cspan+class%3D%22nowrap%22%3E241-%3C%2Fspan%3E254&rft.date=1829&rft_id=info%3Adoi%2F10.1098%2Frstl.1829.0022&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186211638%23id-name%3DS2CID&rft.aulast=Warren&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frstl.1829.0022&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarren1829" class="citation journal cs1">Warren, John (1829). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1829.0031">"On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers"</a>. Philosophical Transactions of the Royal Society of London. 119: 339–359. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1829.0031">10.1098/rstl.1829.0031</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125699726">125699726</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=On+the+geometrical+representation+of+the+powers+of+quantities%2C+whose+indices+involve+the+square+roots+of+negative+numbers&rft.volume=119&rft.pages=%3Cspan+class%3D%22nowrap%22%3E339-%3C%2Fspan%3E359&rft.date=1829&rft_id=info%3Adoi%2F10.1098%2Frstl.1829.0031&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125699726%23id-name%3DS2CID&rft.aulast=Warren&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frstl.1829.0031&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFrançais1813" class="citation journal cs1 cs1-prop-foreign-lang-source">Français, J.F. (1813). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=uc1.$c126478&view=1up&seq=69">"Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires"</a> [New principles of the geometry of position, and geometric interpretation of complex [number] symbols]. Annales des mathématiques pures et appliquées (in French). 4: 61–71.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+des+math%C3%A9matiques+pures+et+appliqu%C3%A9es&rft.atitle=Nouveaux+principes+de+g%C3%A9om%C3%A9trie+de+position%2C+et+interpr%C3%A9tation+g%C3%A9om%C3%A9trique+des+symboles+imaginaires&rft.volume=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E61-%3C%2Fspan%3E71&rft.date=1813&rft.aulast=Fran%C3%A7ais&rft.aufirst=J.F.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Duc1.%24c126478%26view%3D1up%26seq%3D69&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCaparrini2000" class="citation book cs1">Caparrini, Sandro (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139">"On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers"</a>. In Kim Williams (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=voFsJ1EhCnYC">Two Cultures</a>. Birkhäuser. p. 139. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-7186-9" title="Special:BookSources/978-3-7643-7186-9"><bdi>978-3-7643-7186-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+Common+Origin+of+Some+of+the+Works+on+the+Geometrical+Interpretation+of+Complex+Numbers&rft.btitle=Two+Cultures&rft.pages=139&rft.pub=Birkh%C3%A4user&rft.date=2000&rft.isbn=978-3-7643-7186-9&rft.aulast=Caparrini&rft.aufirst=Sandro&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvoFsJ1EhCnYC%26pg%3DPA139&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHardyWright2000" class="citation book cs1">Hardy, G.H.; Wright, E.M. (2000) [1938]. An Introduction to the Theory of Numbers. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">OUP Oxford</a>. p. 189 (fourth edition). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921986-5" title="Special:BookSources/978-0-19-921986-5"><bdi>978-0-19-921986-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.pages=189+%28fourth+edition%29&rft.pub=OUP+Oxford&rft.date=2000&rft.isbn=978-0-19-921986-5&rft.aulast=Hardy&rft.aufirst=G.H.&rft.au=Wright%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJeff_Miller1999" class="citation web cs1 cs1-prop-unfit">Jeff Miller (21 September 1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/19991003034827/http://members.aol.com/jeff570/m.html">"MODULUS"</a>. Earliest Known Uses of Some of the Words of Mathematics (M). Archived from the original on 3 October 1999.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics+%28M%29&rft.atitle=MODULUS&rft.date=1999-09-21&rft.au=Jeff+Miller&rft_id=http%3A%2F%2Fmembers.aol.com%2Fjeff570%2Fm.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCauchy1821" class="citation book cs1 cs1-prop-foreign-lang-source">Cauchy, Augustin-Louis (1821). <a rel="nofollow" class="external text" href="https://archive.org/details/coursdanalysede00caucgoog/page/n209/mode/2up">Cours d'analyse de l'École royale polytechnique</a> (in French). Vol. 1. Paris, France: L'Imprimerie Royale. p. 183.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+d%27analyse+de+l%27%C3%89cole+royale+polytechnique&rft.place=Paris%2C+France&rft.pages=183&rft.pub=L%27Imprimerie+Royale&rft.date=1821&rft.aulast=Cauchy&rft.aufirst=Augustin-Louis&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcoursdanalysede00caucgoog%2Fpage%2Fn209%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <a href="#CITEREFGauss1831">Gauss 1831</a>, p. 96 </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFGauss1831">Gauss 1831</a>, p. 96 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <a href="#CITEREFGauss1831">Gauss 1831</a>, p. 98 </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHankel1867" class="citation book cs1 cs1-prop-foreign-lang-source">Hankel, Hermann (1867). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=754KAAAAYAAJ&pg=PA71">Vorlesungen über die complexen Zahlen und ihre Functionen</a> [Lectures About the Complex Numbers and Their Functions] (in German). Vol. 1. Leipzig, [Germany]: Leopold Voss. p. 71.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+die+complexen+Zahlen+und+ihre+Functionen&rft.place=Leipzig%2C+%5BGermany%5D&rft.pages=71&rft.pub=Leopold+Voss&rft.date=1867&rft.aulast=Hankel&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D754KAAAAYAAJ%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> From p. 71: "Wir werden den Factor (cos φ + i sin φ) haüfig den Richtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".) </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §VIII.1 </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLester1994" class="citation journal cs1">Lester, J.A. (1994). "Triangles I: Shapes". <a href="/wiki/Aequationes_Mathematicae" title="Aequationes Mathematicae">Aequationes Mathematicae</a>. 52: 30–54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01818325">10.1007/BF01818325</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121095307">121095307</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Aequationes+Mathematicae&rft.atitle=Triangles+I%3A+Shapes&rft.volume=52&rft.pages=%3Cspan+class%3D%22nowrap%22%3E30-%3C%2Fspan%3E54&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2FBF01818325&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121095307%23id-name%3DS2CID&rft.aulast=Lester&rft.aufirst=J.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKalman2008a" class="citation journal cs1">Kalman, Dan (2008a). <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1">"An Elementary Proof of Marden's Theorem"</a>. <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 115 (4): 330–38. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2008.11920532">10.1080/00029890.2008.11920532</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13222698">13222698</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1">Archived</a> from the original on 8 March 2012. Retrieved 1 January 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=An+Elementary+Proof+of+Marden%27s+Theorem&rft.volume=115&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E330-%3C%2Fspan%3E38&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13222698%23id-name%3DS2CID&rft.issn=0002-9890&rft_id=info%3Adoi%2F10.1080%2F00029890.2008.11920532&rft.aulast=Kalman&rft.aufirst=Dan&rft_id=http%3A%2F%2Fmathdl.maa.org%2FmathDL%2F22%2F%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D3338%26pf%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKalman2008b" class="citation journal cs1">Kalman, Dan (2008b). <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663">"The Most Marvelous Theorem in Mathematics"</a>. <a href="/wiki/Journal_of_Online_Mathematics_and_Its_Applications" class="mw-redirect" title="Journal of Online Mathematics and Its Applications">Journal of Online Mathematics and Its Applications</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663">Archived</a> from the original on 8 February 2012. Retrieved 1 January 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Online+Mathematics+and+Its+Applications&rft.atitle=The+Most+Marvelous+Theorem+in+Mathematics&rft.date=2008&rft.aulast=Kalman&rft.aufirst=Dan&rft_id=http%3A%2F%2Fmathdl.maa.org%2FmathDL%2F4%2F%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D1663&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrantPhillips2008" class="citation book cs1">Grant, I.S.; Phillips, W.R. (2008). Electromagnetism (2 ed.). Manchester Physics Series. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92712-9" title="Special:BookSources/978-0-471-92712-9"><bdi>978-0-471-92712-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetism&rft.edition=2&rft.pub=Manchester+Physics+Series&rft.date=2008&rft.isbn=978-0-471-92712-9&rft.aulast=Grant&rft.aufirst=I.S.&rft.au=Phillips%2C+W.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMarker1996" class="citation book cs1">Marker, David (1996). <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.lnl/1235423155">"Introduction to the Model Theory of Fields"</a>. In Marker, D.; Messmer, M.; Pillay, A. (eds.). Model theory of fields. Lecture Notes in Logic. Vol. 5. Berlin: Springer-Verlag. pp. 1–37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-60741-0" title="Special:BookSources/978-3-540-60741-0"><bdi>978-3-540-60741-0</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=1477154">1477154</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction+to+the+Model+Theory+of+Fields&rft.btitle=Model+theory+of+fields&rft.place=Berlin&rft.series=Lecture+Notes+in+Logic&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E37&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=978-3-540-60741-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1477154%23id-name%3DMR&rft.aulast=Marker&rft.aufirst=David&rft_id=https%3A%2F%2Fprojecteuclid.org%2Feuclid.lnl%2F1235423155&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBourbaki1998§VIII.4-68"><a href="#cite_ref-FOOTNOTEBourbaki1998§VIII.4_68-0">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §VIII.4. </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcCrimmon2004" class="citation book cs1"><a href="/wiki/Kevin_McCrimmon" title="Kevin McCrimmon">McCrimmon, Kevin</a> (2004). A Taste of Jordan Algebras. Universitext. Springer. p. 64. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95447-3" title="Special:BookSources/0-387-95447-3"><bdi>0-387-95447-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Taste+of+Jordan+Algebras&rft.series=Universitext&rft.pages=64&rft.pub=Springer&rft.date=2004&rft.isbn=0-387-95447-3&rft.aulast=McCrimmon&rft.aufirst=Kevin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2014924">2014924</a> </li> <li id="cite_note-FOOTNOTEApostol198125-70"><a href="#cite_ref-FOOTNOTEApostol198125_70-0">^</a> <a href="#CITEREFApostol1981">Apostol 1981</a>, p. 25. </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAhlfors1979" class="citation book cs1"><a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Ahlfors, Lars</a> (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/lars-ahlfors-complex-analysis-third-edition-mcgraw-hill-science_engineering_math-1979/page/n1/mode/2up">Complex analysis</a> (3rd ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-000657-7" title="Special:BookSources/978-0-07-000657-7"><bdi>978-0-07-000657-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1979&rft.isbn=978-0-07-000657-7&rft.aulast=Ahlfors&rft.aufirst=Lars&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flars-ahlfors-complex-analysis-third-edition-mcgraw-hill-science_engineering_math-1979%2Fpage%2Fn1%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAndreescuAndrica2014" class="citation cs2">Andreescu, Titu; Andrica, Dorin (2014), Complex Numbers from A to ... Z (Second ed.), New York: Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-8415-0">10.1007/978-0-8176-8415-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-8414-3" title="Special:BookSources/978-0-8176-8414-3"><bdi>978-0-8176-8414-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Numbers+from+A+to+...+Z&rft.place=New+York&rft.edition=Second&rft.pub=Springer&rft.date=2014&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-8415-0&rft.isbn=978-0-8176-8414-3&rft.aulast=Andreescu&rft.aufirst=Titu&rft.au=Andrica%2C+Dorin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFApostol1981" class="citation book cs1"><a href="/wiki/Tom_Apostol" class="mw-redirect" title="Tom Apostol">Apostol, Tom</a> (1981). Mathematical analysis. Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+analysis&rft.pub=Addison-Wesley&rft.date=1981&rft.aulast=Apostol&rft.aufirst=Tom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAufmannBarkerNation2007" class="citation book cs1">Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66">College Algebra and Trigonometry</a> (6 ed.). Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-618-82515-8" title="Special:BookSources/978-0-618-82515-8"><bdi>978-0-618-82515-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Algebra+and+Trigonometry&rft.edition=6&rft.pub=Cengage+Learning&rft.date=2007&rft.isbn=978-0-618-82515-8&rft.aulast=Aufmann&rft.aufirst=Richard+N.&rft.au=Barker%2C+Vernon+C.&rft.au=Nation%2C+Richard+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg5j-cT-vg_wC%26pg%3DPA66&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Conway, John B. (1986). Functions of One Complex Variable I. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90328-6" title="Special:BookSources/978-0-387-90328-6"><bdi>978-0-387-90328-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+One+Complex+Variable+I&rft.pub=Springer&rft.date=1986&rft.isbn=978-0-387-90328-6&rft.aulast=Conway&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDerbyshire2006" class="citation book cs1"><a href="/wiki/John_Derbyshire" title="John Derbyshire">Derbyshire, John</a> (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780309096577">Unknown Quantity: A real and imaginary history of algebra</a>. Joseph Henry Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-309-09657-7" title="Special:BookSources/978-0-309-09657-7"><bdi>978-0-309-09657-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unknown+Quantity%3A+A+real+and+imaginary+history+of+algebra&rft.pub=Joseph+Henry+Press&rft.date=2006&rft.isbn=978-0-309-09657-7&rft.aulast=Derbyshire&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780309096577&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Joshi, Kapil D. (1989). Foundations of Discrete Mathematics. New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-21152-6" title="Special:BookSources/978-0-470-21152-6"><bdi>978-0-470-21152-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Discrete+Mathematics&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=978-0-470-21152-6&rft.aulast=Joshi&rft.aufirst=Kapil+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNeedham1997" class="citation book cs1">Needham, Tristan (1997). Visual Complex Analysis. Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853447-1" title="Special:BookSources/978-0-19-853447-1"><bdi>978-0-19-853447-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+Complex+Analysis&rft.pub=Clarendon+Press&rft.date=1997&rft.isbn=978-0-19-853447-1&rft.aulast=Needham&rft.aufirst=Tristan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><a href="/wiki/Daniel_Pedoe" title="Daniel Pedoe">Pedoe, Dan</a> (1988). Geometry: A comprehensive course. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65812-4" title="Special:BookSources/978-0-486-65812-4"><bdi>978-0-486-65812-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+A+comprehensive+course&rft.pub=Dover&rft.date=1988&rft.isbn=978-0-486-65812-4&rft.aulast=Pedoe&rft.aufirst=Dan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPenrose2005" class="citation book cs1"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, Roger</a> (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/roadtorealitycom00penr_0">The Road to Reality: A complete guide to the laws of the universe</a>. Alfred A. Knopf. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-679-45443-4" title="Special:BookSources/978-0-679-45443-4"><bdi>978-0-679-45443-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality%3A+A+complete+guide+to+the+laws+of+the+universe&rft.pub=Alfred+A.+Knopf&rft.date=2005&rft.isbn=978-0-679-45443-4&rft.aulast=Penrose&rft.aufirst=Roger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Froadtorealitycom00penr_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200313111530/http://apps.nrbook.com/empanel/index.html?pg=225">"Section 5.5 Complex Arithmetic"</a>. Numerical Recipes: The art of scientific computing (3rd ed.). New York: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=225">the original</a> on 13 March 2020. Retrieved 9 August 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+5.5+Complex+Arithmetic&rft.btitle=Numerical+Recipes%3A+The+art+of+scientific+computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=W.H.&rft.au=Teukolsky%2C+S.A.&rft.au=Vetterling%2C+W.T.&rft.au=Flannery%2C+B.P.&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%3Fpg%3D225&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSolomentsev2001" class="citation cs2">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Complex_number">"Complex number"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Complex+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Solomentsev&rft.aufirst=E.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DComplex_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3>[<a href="/w/index.php?title=Complex_number&action=edit&section=43" title="Edit section: Historical references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316" /><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArgand1814" class="citation journal cs1 cs1-prop-foreign-lang-source">Argand (1814). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=uc1.$c126479&view=1up&seq=209">"Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise"</a> [Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis]. Annales de mathématiques pures et appliquées (in French). 5: 197–209.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+math%C3%A9matiques+pures+et+appliqu%C3%A9es&rft.atitle=Reflexions+sur+la+nouvelle+th%C3%A9orie+des+imaginaires%2C+suives+d%27une+application+%C3%A0+la+demonstration+d%27un+theor%C3%A8me+d%27analise&rft.volume=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E209&rft.date=1814&rft.au=Argand&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Duc1.%24c126479%26view%3D1up%26seq%3D209&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1998). "Foundations of mathematics § logic: set theory". Elements of the history of mathematics. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foundations+of+mathematics+%C2%A7+logic%3A+set+theory&rft.btitle=Elements+of+the+history+of+mathematics&rft.pub=Springer&rft.date=1998&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Burton, David M. (1995). The History of Mathematics (3rd ed.). New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-009465-9" title="Special:BookSources/978-0-07-009465-9"><bdi>978-0-07-009465-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematics&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1995&rft.isbn=978-0-07-009465-9&rft.aulast=Burton&rft.aufirst=David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGauss1831" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, C. F.</a> (1831). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=283">"Theoria residuorum biquadraticorum. Commentatio secunda"</a> [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commentationes+Societatis+Regiae+Scientiarum+Gottingensis+Recentiores&rft.atitle=Theoria+residuorum+biquadraticorum.+Commentatio+secunda.&rft.volume=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E89-%3C%2Fspan%3E148&rft.date=1831&rft.aulast=Gauss&rft.aufirst=C.+F.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015073697180%26view%3D1up%26seq%3D283&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Katz, Victor J. (2004). A History of Mathematics, Brief Version. <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-16193-2" title="Special:BookSources/978-0-321-16193-2"><bdi>978-0-321-16193-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%2C+Brief+Version&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-321-16193-2&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Nahin, Paul J. (1998). An Imaginary Tale: The Story of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba09297ec8ad80d38116c988c033ae42e0d03ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.469ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\sqrt {-1}}}" />. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02795-1" title="Special:BookSources/978-0-691-02795-1"><bdi>978-0-691-02795-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Imaginary+Tale%3A+The+Story+of+MATH+RENDER+ERROR&rft.pub=Princeton+University+Press&rft.date=1998&rft.isbn=978-0-691-02795-1&rft.aulast=Nahin&rft.aufirst=Paul+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEbbinghausHermesHirzebruchKoecher1991" class="citation book cs1">Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991). Numbers (hardcover ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97497-2" title="Special:BookSources/978-0-387-97497-2"><bdi>978-0-387-97497-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers&rft.edition=hardcover&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-97497-2&rft.aulast=Ebbinghaus&rft.aufirst=H.+D.&rft.au=Hermes%2C+H.&rft.au=Hirzebruch%2C+F.&rft.au=Koecher%2C+M.&rft.au=Mainzer%2C+K.&rft.au=Neukirch%2C+J.&rft.au=Prestel%2C+A.&rft.au=Remmert%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+number" class="Z3988"> — An advanced perspective on the historical development of the concept of number.</li></ul> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Complex_numbers19" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_numbers" title="Template:Complex numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complex_numbers" title="Template talk:Complex numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complex_numbers" title="Special:EditPage/Template:Complex numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Complex_numbers19" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Complex numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complex_conjugate" title="Complex conjugate">Complex conjugate</a></li> <li><a href="/wiki/Complex_plane" title="Complex plane">Complex plane</a></li> <li><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary number</a></li> <li><a href="/wiki/Real_number" title="Real number">Real number</a></li> <li><a href="/wiki/Unit_complex_number" class="mw-redirect" title="Unit complex number">Unit complex number</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Number_systems351" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems351" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<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><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} }" />)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }" />)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" />)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />)</li> <li><a class="mw-selflink selflink">Complex numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" />)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }" />)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a> Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" />)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }" />)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)">p-adic solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1551" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q11567#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4128698-4">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85093211">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11981946j">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11981946j">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00563643">Japan</a></li><li><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph121761&CON_LNG=ENG">Czech Republic</a></li><li><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000082623&P_CON_LNG=ENG">Latvia</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007538749605171">Israel</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Complex_number&oldid=1281498490">https://en.wikipedia.org/w/index.php?title=Complex_number&oldid=1281498490</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:Composition_algebras" title="Category:Composition algebras">Composition algebras</a></li><li><a href="/wiki/Category:Complex_numbers" title="Category:Complex numbers">Complex numbers</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_Latin-language_sources_(la)" title="Category:CS1 Latin-language sources (la)">CS1 Latin-language sources (la)</a></li><li><a href="/wiki/Category:CS1_Danish-language_sources_(da)" title="Category:CS1 Danish-language sources (da)">CS1 Danish-language sources (da)</a></li><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:CS1:_unfit_URL" title="Category:CS1: unfit URL">CS1: unfit URL</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_indefinitely_move-protected_pages" title="Category:Wikipedia indefinitely move-protected pages">Wikipedia indefinitely move-protected pages</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_June_2020" title="Category:Use dmy dates from June 2020">Use dmy dates from June 2020</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li><li><a href="/wiki/Category:Pages_that_use_a_deprecated_format_of_the_math_tags" title="Category:Pages that use a deprecated format of the math tags">Pages that use a deprecated format of the math tags</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 20 March 2025, at 19:00 (UTC).</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=Complex_number&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"> Search </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" >Complex number</div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"> 132 languages </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"> Add topic </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.eqiad.main-d6f4c4464-hhhlx","wgBackendResponseTime":375,"wgPageParseReport":{"limitreport":{"cputime":"2.101","walltime":"2.701","ppvisitednodes":{"value":12976,"limit":1000000},"postexpandincludesize":{"value":191514,"limit":2097152},"templateargumentsize":{"value":18285,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":268751,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1817.259 1 -total"," 28.77% 522.915 2 Template:Reflist"," 25.45% 462.443 47 Template:Cite_book"," 9.76% 177.446 106 Template:Math"," 9.39% 170.676 1 Template:Short_description"," 7.03% 127.672 2 Template:Navbox"," 6.64% 120.742 1 Template:Complex_numbers"," 6.28% 114.114 2 Template:Pagetype"," 6.10% 110.916 12 Template:Harvnb"," 6.09% 110.636 8 Template:Efn"]},"scribunto":{"limitreport-timeusage":{"value":"1.015","limit":"10.000"},"limitreport-memusage":{"value":8605407,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAhlfors1979\"] = 1,\n [\"CITEREFAndreescuAndrica2014\"] = 1,\n [\"CITEREFApostol1981\"] = 1,\n [\"CITEREFArgand1806\"] = 1,\n [\"CITEREFArgand1814\"] = 1,\n [\"CITEREFAufmannBarkerNation2007\"] = 1,\n [\"CITEREFAxler2010\"] = 1,\n [\"CITEREFBourbaki1998\"] = 1,\n [\"CITEREFBrownChurchill1996\"] = 1,\n [\"CITEREFBryan_Bunch2012\"] = 1,\n [\"CITEREFBuée1806\"] = 1,\n [\"CITEREFCampbell1911\"] = 1,\n [\"CITEREFCaparrini2000\"] = 1,\n [\"CITEREFCauchy1821\"] = 1,\n [\"CITEREFConfalonieri2015\"] = 1,\n [\"CITEREFCynthia_Y._Young2017\"] = 1,\n [\"CITEREFDennis_ZillJacqueline_Dewar2011\"] = 1,\n [\"CITEREFDerbyshire2006\"] = 1,\n [\"CITEREFDescartes1954\"] = 1,\n [\"CITEREFEbbinghausHermesHirzebruchKoecher1991\"] = 1,\n [\"CITEREFEuler1748\"] = 1,\n [\"CITEREFEwald1996\"] = 1,\n [\"CITEREFFrançais1813\"] = 1,\n [\"CITEREFGauss1831\"] = 1,\n [\"CITEREFGrantPhillips2008\"] = 1,\n [\"CITEREFHamilton1844\"] = 1,\n [\"CITEREFHankel1867\"] = 1,\n [\"CITEREFHardyWright2000\"] = 1,\n [\"CITEREFJeff_Miller1999\"] = 1,\n [\"CITEREFJoseph_Mazur2016\"] = 1,\n [\"CITEREFJurij.\"] = 1,\n [\"CITEREFKalman2008a\"] = 1,\n [\"CITEREFKalman2008b\"] = 1,\n [\"CITEREFKasana2005\"] = 1,\n [\"CITEREFKatz2004\"] = 1,\n [\"CITEREFKline\"] = 1,\n [\"CITEREFLester1994\"] = 1,\n [\"CITEREFLloyd_James_Peter_Kilford2015\"] = 1,\n [\"CITEREFMarker1996\"] = 1,\n [\"CITEREFMcCrimmon2004\"] = 1,\n [\"CITEREFMourey1861\"] = 1,\n [\"CITEREFNahin2007\"] = 1,\n [\"CITEREFNeedham1997\"] = 1,\n [\"CITEREFNilssonRiedel2008\"] = 1,\n [\"CITEREFPedoe1988\"] = 1,\n [\"CITEREFPenrose2005\"] = 1,\n [\"CITEREFPressTeukolskyVetterlingFlannery2007\"] = 1,\n [\"CITEREFSolomentsev2001\"] = 1,\n [\"CITEREFSpiegelLipschutzSchillerSpellman2009\"] = 1,\n [\"CITEREFWallis1685\"] = 1,\n [\"CITEREFWarren1828\"] = 1,\n [\"CITEREFWarren1829\"] = 2,\n [\"CITEREFWeisstein\"] = 1,\n [\"CITEREFWessel1799\"] = 1,\n [\"CITEREFWilliam_Ford2014\"] = 1,\n [\"Multiplication\"] = 1,\n [\"Polar_form\"] = 1,\n [\"Square\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 3,\n [\"-2-2i\"] = 1,\n [\"Anchor\"] = 2,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite book\"] = 47,\n [\"Cite journal\"] = 12,\n [\"Cite web\"] = 3,\n [\"Commons category\"] = 1,\n [\"Complex numbers\"] = 1,\n [\"DEFAULTSORT:Complex Number\"] = 1,\n [\"EB1911 poster\"] = 1,\n [\"Efn\"] = 8,\n [\"Harvnb\"] = 12,\n [\"Main\"] = 5,\n [\"Math\"] = 101,\n [\"Mr\"] = 1,\n [\"Mvar\"] = 94,\n [\"No\"] = 11,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 4,\n [\"Number systems\"] = 1,\n [\"Open-closed\"] = 1,\n [\"Overline\"] = 2,\n [\"Pi\"] = 1,\n [\"Pipe\"] = 1,\n [\"Pp-move\"] = 1,\n [\"Redirect\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"See also\"] = 3,\n [\"Sfn\"] = 5,\n [\"Sfrac\"] = 1,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Sup\"] = 2,\n [\"TOC limit\"] = 1,\n [\"Tmath\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Visible anchor\"] = 2,\n [\"Visualisation complex number roots\"] = 1,\n [\"Wikibooks\"] = 1,\n [\"Wikiversity\"] = 1,\n [\"Yes\"] = 18,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n","limitreport-profile":[["?","260","26.5"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","180","18.4"],["dataWrapper \u003Cmw.lua:672\u003E","160","16.3"],["recursiveClone \u003CmwInit.lua:45\u003E","60","6.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","60","6.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","40","4.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::anchorEncode","40","4.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","20","2.0"],["assert","20","2.0"],["? \u003CModule:Citation/CS1/COinS:160\u003E","20","2.0"],["[others]","120","12.2"]]},"cachereport":{"origin":"mw-web.eqiad.main-65bf7dbd64-f2rlr","timestamp":"20250325193050","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Complex number","url":"https:\/\/en.wikipedia.org\/wiki\/Complex_number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q11567","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q11567","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-11-18T22:01:51Z","dateModified":"2025-03-20T19:00:22Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/50\/A_plus_bi.svg","headline":"number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit"}</script> </body> </html>