CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="ast" dir="ltr"> <head> <meta charset="UTF-8"> <title>Númberu real - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )astwikimwclientpreferences=([^;]+)/);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":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","xineru","febreru","marzu","abril","mayu","xunu","xunetu","agostu","setiembre","ochobre","payares","avientu"],"wgRequestId":"5973e9a5-2f13-42f6-8a50-1f685126229c","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Númberu_real","wgTitle":"Númberu real","wgCurRevisionId":4178560,"wgRevisionId":4178560,"wgArticleId":138067,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikipedia:Artículos qu'usen ficha ensin datos en fileres","Wikipedia:Artículos con datos ensin referencies","Wikipedia:Páxines con enllaz commonscat igual que Wikidata","Wikipedia:Artículos con identificadores BNF","Wikipedia:Artículos con identificadores GND","Wikipedia:Artículos con identificadores LCCN","Wikipedia:Tradubot","Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten","Wikipedia:Revisar traducción" ,"Númberos reales","Matemática elemental"],"wgPageViewLanguage":"ast","wgPageContentLanguage":"ast","wgPageContentModel":"wikitext","wgRelevantPageName":"Númberu_real","wgRelevantArticleId":138067,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"ast","pageLanguageDir":"ltr","pageVariantFallbacks":"ast"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q12916","wgCheckUserClientHintsHeadersJsApi":[ "brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.urlShortener.toolbar", "ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=ast&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=ast&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=ast&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/1200px-N%C3%BAmeros_reales.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1329"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/800px-N%C3%BAmeros_reales.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="886"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/640px-N%C3%BAmeros_reales.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="709"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Númberu real - 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="//ast.m.wikipedia.org/wiki/N%C3%BAmberu_real"> <link rel="alternate" type="application/x-wiki" title="Editar" href="/w/index.php?title=N%C3%BAmberu_real&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 (ast)"> <link rel="EditURI" type="application/rsd+xml" href="//ast.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_real"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ast"> <link rel="alternate" type="application/atom+xml" title="Canal Atom Wikipedia" href="/w/index.php?title=Especial:CambeosRecientes&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-Númberu_real rootpage-Númberu_real skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Saltar al conteníu</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="Sitio"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">despintar</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegación </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visita la portada [z]" accesskey="z"><span>Portada</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Portal_de_la_comunid%C3%A1" title="Tocante al proyeutu, lo qué pues facer, ú s'alcuentren les coses"><span>Portal de la comunidá</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:CambeosRecientes" title="La llista de cambios recientes de la wiki. [r]" accesskey="r"><span>Cambeos recién</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Aleatoria" title="Carga una páxina al debalu [x]" accesskey="x"><span>Páxina al debalu</span></a></li><li id="n-help" class="mw-list-item"><a href="https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents" title="El llugar pa deprender"><span>Ayuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="La Enciclopedia Llibre" src="/static/images/mobile/copyright/wikipedia-tagline-ast.svg" width="119" height="13" style="width: 7.4375em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Gueta" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Busca en Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Buscar</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Buscar en Wikipedia" aria-label="Buscar en Wikipedia" autocapitalize="sentences" title="Busca en Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Gueta"> </div> <button class="cdx-button cdx-search-input__end-button">Guetar</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Ferramientes personales"> <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="Apariencia"> <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="Apariencia" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Apariencia</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ast.wikipedia.org&uselang=ast" class=""><span>Donativos</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crear_una_cuenta&returnto=N%C3%BAmberu+real" title="Encamentámoste que crees una cuenta y qu'anicies sesión; sicasí, nun ye obligatorio" class=""><span>Crear una cuenta</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Entrar&returnto=N%C3%BAmberu+real" title="T'encamentamos que t'identifiques, anque nun ye obligatorio [o]" accesskey="o" class=""><span>Entrar</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Más opciones" > <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="Ferramientes personales" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Ferramientes personales</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú de usuario" > <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=ast.wikipedia.org&uselang=ast"><span>Donativos</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_una_cuenta&returnto=N%C3%BAmberu+real" title="Encamentámoste que crees una cuenta y qu'anicies sesión; sicasí, nun ye obligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear una cuenta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Entrar&returnto=N%C3%BAmberu+real" title="T'encamentamos que t'identifiques, anque nun ye obligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Entrar</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Páginas para editores desconectados <a href="/wiki/Ayuda:Introducci%C3%B3n" aria-label="Obtenga más información sobre editar"><span>más información</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:MisContribuciones" title="Una llista d'ediciones feches dende esta dirección IP [y]" accesskey="y"><span>Contribuciones</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:MiDiscusi%C3%B3n" title="Alderique de les ediciones feches con esta direición IP [n]" accesskey="n"><span>Alderique</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sitio"> <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="Conteníu" 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">Conteníu</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">despintar</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">Entamu</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <button aria-controls="toc-Historia-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Historia</span> </button> <ul id="toc-Historia-sublist" class="vector-toc-list"> <li id="toc-Evolución_del_conceutu_de_númberu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evolución_del_conceutu_de_númberu"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Evolución del conceutu de númberu</span> </div> </a> <ul id="toc-Evolución_del_conceutu_de_númberu-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notación" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notación"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Notación</span> </div> </a> <ul id="toc-Notación-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tipos_de_númberos_reales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tipos_de_númberos_reales"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tipos de númberos reales</span> </div> </a> <button aria-controls="toc-Tipos_de_númberos_reales-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Tipos de númberos reales</span> </button> <ul id="toc-Tipos_de_númberos_reales-sublist" class="vector-toc-list"> <li id="toc-Racionales_ya_irracionales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racionales_ya_irracionales"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Racionales ya irracionales</span> </div> </a> <ul id="toc-Racionales_ya_irracionales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alxebraicos_y_trescendentes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alxebraicos_y_trescendentes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Alxebraicos y trescendentes</span> </div> </a> <ul id="toc-Alxebraicos_y_trescendentes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computables_y_irreductibles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computables_y_irreductibles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Computables y irreductibles</span> </div> </a> <ul id="toc-Computables_y_irreductibles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Construcciones_del_conxuntu_de_númberos_reales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construcciones_del_conxuntu_de_númberos_reales"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Construcciones del conxuntu de númberos reales</span> </div> </a> <button aria-controls="toc-Construcciones_del_conxuntu_de_númberos_reales-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Construcciones del conxuntu de númberos reales</span> </button> <ul id="toc-Construcciones_del_conxuntu_de_númberos_reales-sublist" class="vector-toc-list"> <li id="toc-Presentación_axomática" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Presentación_axomática"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Presentación axomática</span> </div> </a> <ul id="toc-Presentación_axomática-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construcción_por_númberos_decimales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construcción_por_númberos_decimales"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Construcción por númberos decimales</span> </div> </a> <ul id="toc-Construcción_por_númberos_decimales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construcción_por_cortadures_de_Dedekind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construcción_por_cortadures_de_Dedekind"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Construcción por cortadures de Dedekind</span> </div> </a> <ul id="toc-Construcción_por_cortadures_de_Dedekind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construcción_por_socesión_de_Cauchy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construcción_por_socesión_de_Cauchy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Construcción por socesión de Cauchy</span> </div> </a> <ul id="toc-Construcción_por_socesión_de_Cauchy-sublist" class="vector-toc-list"> <li id="toc-Definición_de_los_númberos_reales" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definición_de_los_númberos_reales"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Definición de los númberos reales</span> </div> </a> <ul id="toc-Definición_de_los_númberos_reales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propiedá_Arquimediana_(Axoma_de_Arquímedes)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Propiedá_Arquimediana_(Axoma_de_Arquímedes)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.2</span> <span>Propiedá Arquimediana (Axoma de Arquímedes)</span> </div> </a> <ul id="toc-Propiedá_Arquimediana_(Axoma_de_Arquímedes)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Operaciones_con_númberos_reales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operaciones_con_númberos_reales"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Operaciones con númberos reales</span> </div> </a> <ul id="toc-Operaciones_con_númberos_reales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dos_particiones" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dos_particiones"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Dos particiones</span> </div> </a> <ul id="toc-Dos_particiones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Ver tamién</span> </div> </a> <button aria-controls="toc-Ver_tamién-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Ver tamién</span> </button> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> <li id="toc-Dos_clasificaciones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dos_clasificaciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Dos clasificaciones</span> </div> </a> <ul id="toc-Dos_clasificaciones-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes_y_referencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_y_referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes y referencies</span> </div> </a> <ul id="toc-Notes_y_referencies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaces_esternos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaces_esternos"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Enllaces esternos</span> </div> </a> <ul id="toc-Enllaces_esternos-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteníu" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Númberu real</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 118 idiomas" > <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-118" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">118 llingües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Re%C3%ABle_getal" title="Reële getal – afrikaans" lang="af" hreflang="af" data-title="Reële getal" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – alemán de Suiza" lang="gsw" hreflang="gsw" data-title="Reelle Zahl" data-language-autonym="Alemannisch" data-language-local-name="alemán de Suiza" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%D9%8A%D9%82%D9%8A" title="عدد حقيقي – árabe" lang="ar" hreflang="ar" data-title="عدد حقيقي" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99qiqi_%C9%99d%C9%99dl%C9%99r" title="Həqiqi ədədlər – azerbaixanu" lang="az" hreflang="az" data-title="Həqiqi ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaixanu" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="حقیقی ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="حقیقی ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD" title="Ысын һан – bashkir" lang="ba" hreflang="ba" data-title="Ысын һан" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Tunay na bilang" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – bielorrusu" lang="be" hreflang="be" data-title="Рэчаісны лік" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Рэчаісны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Реално число – búlgaru" lang="bg" hreflang="bg" data-title="Реално число" data-language-autonym="Български" data-language-local-name="búlgaru" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – Bhojpuri" lang="bh" hreflang="bh" data-title="वास्तविक संख्या" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A6%AC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বাস্তব সংখ্যা – bengalín" lang="bn" hreflang="bn" data-title="বাস্তব সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bengalín" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Realan_broj" title="Realan broj – bosniu" lang="bs" hreflang="bs" data-title="Realan broj" data-language-autonym="Bosanski" data-language-local-name="bosniu" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%91%D0%BE%D0%B4%D0%BE%D1%82%D0%BE_%D1%82%D0%BE%D0%BE" title="Бодото тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Бодото тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="artículo bueno"><a href="https://ca.wikipedia.org/wiki/Nombre_real" title="Nombre real – catalán" lang="ca" hreflang="ca" data-title="Nombre real" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D9%82%DB%8C%D9%86%DB%95" title="ژمارەی ڕاستەقینە – kurdu central" lang="ckb" hreflang="ckb" data-title="ژمارەی ڕاستەقینە" data-language-autonym="کوردی" data-language-local-name="kurdu central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Aqiqiy_say%C4%B1" title="Aqiqiy sayı – turcu de Crimea" lang="crh" hreflang="crh" data-title="Aqiqiy sayı" data-language-autonym="Qırımtatarca" data-language-local-name="turcu de Crimea" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo – checu" lang="cs" hreflang="cs" data-title="Reálné číslo" data-language-autonym="Čeština" data-language-local-name="checu" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A7%C4%83%D0%BD_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Чăн хисеп – chuvash" lang="cv" hreflang="cv" data-title="Чăн хисеп" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_real" title="Rhif real – galés" lang="cy" hreflang="cy" data-title="Rhif real" data-language-autonym="Cymraeg" data-language-local-name="galés" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reelle_tal" title="Reelle tal – danés" lang="da" hreflang="da" data-title="Reelle tal" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – alemán" lang="de" hreflang="de" data-title="Reelle Zahl" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_reel" title="Amaro reel – Zazaki" lang="diq" hreflang="diq" data-title="Amaro reel" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πραγματικός αριθμός – griegu" lang="el" hreflang="el" data-title="Πραγματικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="griegu" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_re%C3%A8l" title="Nómmer reèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer reèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Real_number" title="Real number – inglés" lang="en" hreflang="en" data-title="Real number" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Reelo" title="Reelo – esperanto" lang="eo" hreflang="eo" data-title="Reelo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – español" lang="es" hreflang="es" data-title="Número real" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Reaalarv" title="Reaalarv – estoniu" lang="et" hreflang="et" data-title="Reaalarv" data-language-autonym="Eesti" data-language-local-name="estoniu" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_erreal" title="Zenbaki erreal – vascu" lang="eu" hreflang="eu" data-title="Zenbaki erreal" data-language-autonym="Euskara" data-language-local-name="vascu" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%DB%8C%D9%82%DB%8C" title="عدد حقیقی – persa" lang="fa" hreflang="fa" data-title="عدد حقیقی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reaaliluku" title="Reaaliluku – finlandés" lang="fi" hreflang="fi" data-title="Reaaliluku" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Reaalarv" title="Reaalarv – voro" lang="vro" hreflang="vro" data-title="Reaalarv" data-language-autonym="Võro" data-language-local-name="voro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Reelt_tal" title="Reelt tal – feroés" lang="fo" hreflang="fo" data-title="Reelt tal" data-language-autonym="Føroyskt" data-language-local-name="feroés" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_r%C3%A9el" title="Nombre réel – francés" lang="fr" hreflang="fr" data-title="Nombre réel" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Re%27el_taal" title="Re'el taal – frisón del norte" lang="frr" hreflang="frr" data-title="Re'el taal" data-language-autonym="Nordfriisk" data-language-local-name="frisón del norte" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Numars_re%C3%A2i" title="Numars reâi – friulianu" lang="fur" hreflang="fur" data-title="Numars reâi" data-language-autonym="Furlan" data-language-local-name="friulianu" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9aduimhir" title="Réaduimhir – irlandés" lang="ga" hreflang="ga" data-title="Réaduimhir" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – chinu gan" lang="gan" hreflang="gan" data-title="實數" data-language-autonym="贛語" data-language-local-name="chinu gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_r%C3%A9y%C3%A8l" title="Nonm réyèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm réyèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – gallegu" lang="gl" hreflang="gl" data-title="Número real" data-language-autonym="Galego" data-language-local-name="gallegu" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Feer_earroo" title="Feer earroo – manés" lang="gv" hreflang="gv" data-title="Feer earroo" data-language-autonym="Gaelg" data-language-local-name="manés" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי – hebréu" lang="he" hreflang="he" data-title="מספר ממשי" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – hindi" lang="hi" hreflang="hi" data-title="वास्तविक संख्या" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Realni_broj" title="Realni broj – croata" lang="hr" hreflang="hr" data-title="Realni broj" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok – húngaru" lang="hu" hreflang="hu" data-title="Valós számok" data-language-autonym="Magyar" data-language-local-name="húngaru" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Իրական թիվ – armeniu" lang="hy" hreflang="hy" data-title="Իրական թիվ" data-language-autonym="Հայերեն" data-language-local-name="armeniu" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_real" title="Numero real – interlingua" lang="ia" hreflang="ia" data-title="Numero real" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_bendar" title="Lumur bendar – iban" lang="iba" hreflang="iba" data-title="Lumur bendar" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_riil" title="Bilangan riil – indonesiu" lang="id" hreflang="id" data-title="Bilangan riil" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiu" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Reala_nombro" title="Reala nombro – ido" lang="io" hreflang="io" data-title="Reala nombro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Rauntala" title="Rauntala – islandés" lang="is" hreflang="is" data-title="Rauntala" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_reale" title="Numero reale – italianu" lang="it" hreflang="it" data-title="Numero reale" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%9F%E6%95%B0" title="実数 – xaponés" lang="ja" hreflang="ja" data-title="実数" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Riil_nomba" title="Riil nomba – inglés criollu xamaicanu" lang="jam" hreflang="jam" data-title="Riil nomba" data-language-autonym="Patois" data-language-local-name="inglés criollu xamaicanu" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/pavycimdyna%27u" title="pavycimdyna'u – lojban" lang="jbo" hreflang="jbo" data-title="pavycimdyna'u" data-language-autonym="La .lojban." data-language-local-name="lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%9B%E1%83%93%E1%83%95%E1%83%98%E1%83%9A%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ნამდვილი რიცხვი – xeorxanu" lang="ka" hreflang="ka" data-title="ნამდვილი რიცხვი" data-language-autonym="ქართული" data-language-local-name="xeorxanu" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Si%C5%8B%C5%8B_%C3%B1%CA%8A%C5%8B_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Siŋŋ ñʊŋ (tʊʊzʊʊ) – Kabiye" lang="kbp" hreflang="kbp" data-title="Siŋŋ ñʊŋ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D2%9B%D1%82%D1%8B_%D1%81%D0%B0%D0%BD" title="Нақты сан – kazaquistanín" lang="kk" hreflang="kk" data-title="Нақты сан" data-language-autonym="Қазақша" data-language-local-name="kazaquistanín" class="interlanguage-link-target"><span>Қазақша</span></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%96%E1%9E%B7%E1%9E%8F" title="ចំនួនពិត – ḥemer" lang="km" hreflang="km" data-title="ចំនួនពិត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="ḥemer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B3%88%E0%B2%9C_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ನೈಜ ಸಂಖ್ಯೆ – canarés" lang="kn" hreflang="kn" data-title="ನೈಜ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarés" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%A4%EC%88%98" title="실수 – coreanu" lang="ko" hreflang="ko" data-title="실수" data-language-autonym="한국어" data-language-local-name="coreanu" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_rast%C3%AEn" title="Hejmarên rastîn – curdu" lang="ku" hreflang="ku" data-title="Hejmarên rastîn" data-language-autonym="Kurdî" data-language-local-name="curdu" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Анык сан – kirguistanín" lang="ky" hreflang="ky" data-title="Анык сан" data-language-autonym="Кыргызча" data-language-local-name="kirguistanín" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_realis" title="Numerus realis – llatín" lang="la" hreflang="la" data-title="Numerus realis" data-language-autonym="Latina" data-language-local-name="llatín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_real" title="Numero real – lingua franca nova" lang="lfn" hreflang="lfn" data-title="Numero real" data-language-autonym="Lingua Franca Nova" data-language-local-name="lingua franca nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – limburgués" lang="li" hreflang="li" data-title="Reëel getal" data-language-autonym="Limburgs" data-language-local-name="limburgués" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Numeri_re%C3%A6" title="Numeri reæ – ligurianu" lang="lij" hreflang="lij" data-title="Numeri reæ" data-language-autonym="Ligure" data-language-local-name="ligurianu" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_real" title="Numer real – lombardu" lang="lmo" hreflang="lmo" data-title="Numer real" data-language-autonym="Lombard" data-language-local-name="lombardu" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%88%E0%BA%B4%E0%BA%87" title="ຈຳນວນຈິງ – laosianu" lang="lo" hreflang="lo" data-title="ຈຳນວນຈິງ" data-language-autonym="ລາວ" data-language-local-name="laosianu" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Realusis_skai%C4%8Dius" title="Realusis skaičius – lituanu" lang="lt" hreflang="lt" data-title="Realusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="lituanu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Re%C4%81ls_skaitlis" title="Reāls skaitlis – letón" lang="lv" hreflang="lv" data-title="Reāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_voatsapa" title="Isa voatsapa – malgaxe" lang="mg" hreflang="mg" data-title="Isa voatsapa" data-language-autonym="Malagasy" data-language-local-name="malgaxe" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реален број – macedoniu" lang="mk" hreflang="mk" data-title="Реален број" data-language-autonym="Македонски" data-language-local-name="macedoniu" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B4%B5%E0%B4%BF%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="വാസ്തവികസംഖ്യ – malayalam" lang="ml" hreflang="ml" data-title="വാസ്തവികസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – marathi" lang="mr" hreflang="mr" data-title="वास्तविक संख्या" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_nyata" title="Nombor nyata – malayu" lang="ms" hreflang="ms" data-title="Nombor nyata" data-language-autonym="Bahasa Melayu" data-language-local-name="malayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%85%E1%80%85%E1%80%BA" title="ကိန်းစစ် – birmanu" lang="my" hreflang="my" data-title="ကိန်းစစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanu" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक सङ्ख्या – nepalés" lang="ne" hreflang="ne" data-title="वास्तविक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="nepalés" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – neerlandés" lang="nl" hreflang="nl" data-title="Reëel getal" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Reelle_tal" title="Reelle tal – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Reelle tal" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Reelt_tall" title="Reelt tall – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Reelt tall" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_real" title="Nombre real – occitanu" lang="oc" hreflang="oc" data-title="Nombre real" data-language-autonym="Occitan" data-language-local-name="occitanu" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%91%C3%A6%D0%BB%D0%B2%D1%8B%D1%80%D0%B4_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Бæлвырд нымæц – oséticu" lang="os" hreflang="os" data-title="Бæлвырд нымæц" data-language-autonym="Ирон" data-language-local-name="oséticu" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B5%E0%A8%BF%E0%A8%95_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਵਾਸਤਵਿਕ ਅੰਕ – punyabí" lang="pa" hreflang="pa" data-title="ਵਾਸਤਵਿਕ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punyabí" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_rzeczywiste" title="Liczby rzeczywiste – polacu" lang="pl" hreflang="pl" data-title="Liczby rzeczywiste" data-language-autonym="Polski" data-language-local-name="polacu" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_real" title="Nùmer real – piamontés" lang="pms" hreflang="pms" data-title="Nùmer real" data-language-autonym="Piemontèis" data-language-local-name="piamontés" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – portugués" lang="pt" hreflang="pt" data-title="Número real" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_real" title="Număr real – rumanu" lang="ro" hreflang="ro" data-title="Număr real" data-language-autonym="Română" data-language-local-name="rumanu" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вещественное число – rusu" lang="ru" hreflang="ru" data-title="Вещественное число" data-language-autonym="Русский" data-language-local-name="rusu" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D1%8C%D0%B8%D2%A5%D0%BD%D1%8D%D1%8D%D1%85_%D1%87%D1%8B%D1%8B%D2%BB%D1%8B%D0%BB%D0%B0%D0%BB%D0%B0%D1%80" title="Дьиҥнээх чыыһылалар – sakha" lang="sah" hreflang="sah" data-title="Дьиҥнээх чыыһылалар" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_riali" title="Nùmmuru riali – sicilianu" lang="scn" hreflang="scn" data-title="Nùmmuru riali" data-language-autonym="Sicilianu" data-language-local-name="sicilianu" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Realan_broj" title="Realan broj – serbo-croata" lang="sh" hreflang="sh" data-title="Realan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="තාත්වික සංඛ්‍යා – cingalés" lang="si" hreflang="si" data-title="තාත්වික සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="cingalés" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Real_number" title="Real number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Real number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Re%C3%A1lne_%C4%8D%C3%ADslo" title="Reálne číslo – eslovacu" lang="sk" hreflang="sk" data-title="Reálne číslo" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Realno_%C5%A1tevilo" title="Realno število – eslovenu" lang="sl" hreflang="sl" data-title="Realno število" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Reaalloho" title="Reaalloho – inari sami" lang="smn" hreflang="smn" data-title="Reaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="inari sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_real%C3%AB" title="Numrat realë – albanu" lang="sq" hreflang="sq" data-title="Numrat realë" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реалан број – serbiu" lang="sr" hreflang="sr" data-title="Реалан број" data-language-autonym="Српски / srpski" data-language-local-name="serbiu" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reella_tal" title="Reella tal – suecu" lang="sv" hreflang="sv" data-title="Reella tal" data-language-autonym="Svenska" data-language-local-name="suecu" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_halisi" title="Namba halisi – suaḥili" lang="sw" hreflang="sw" data-title="Namba halisi" data-language-autonym="Kiswahili" data-language-local-name="suaḥili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%AF%E0%AF%8D%E0%AE%AF%E0%AF%86%E0%AE%A3%E0%AF%8D" title="மெய்யெண் – tamil" lang="ta" hreflang="ta" data-title="மெய்யெண்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%88%E0%B8%A3%E0%B8%B4%E0%B8%87" title="จำนวนจริง – tailandés" lang="th" hreflang="th" data-title="จำนวนจริง" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – tagalog" lang="tl" hreflang="tl" data-title="Tunay na bilang" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Reel_say%C4%B1lar" title="Reel sayılar – turcu" lang="tr" hreflang="tr" data-title="Reel sayılar" data-language-autonym="Türkçe" data-language-local-name="turcu" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B9%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Дійсне число – ucraín" lang="uk" hreflang="uk" data-title="Дійсне число" data-language-autonym="Українська" data-language-local-name="ucraín" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B9%D8%AF%D8%AF" title="حقیقی عدد – urdu" lang="ur" hreflang="ur" data-title="حقیقی عدد" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Haqiqiy_sonlar" title="Haqiqiy sonlar – uzbequistanín" lang="uz" hreflang="uz" data-title="Haqiqiy sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbequistanín" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_th%E1%BB%B1c" title="Số thực – vietnamín" lang="vi" hreflang="vi" data-title="Số thực" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamín" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – chinu wu" lang="wuu" hreflang="wuu" data-title="实数" data-language-autonym="吴语" data-language-local-name="chinu wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D3%99%D3%99%D0%BB%D2%BB%D0%B0%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Бәәлһан тойг – calmuco" lang="xal" hreflang="xal" data-title="Бәәлһан тойг" data-language-autonym="Хальмг" data-language-local-name="calmuco" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A8%D7%A2%D7%90%D7%9C%D7%A2_%D7%A6%D7%90%D7%9C" title="רעאלע צאל – yiddish" lang="yi" hreflang="yi" data-title="רעאלע צאל" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="artículo bueno"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_gidi" title="Nọ́mbà gidi – yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà gidi" data-language-autonym="Yorùbá" data-language-local-name="yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – chinu" lang="zh" hreflang="zh" data-title="实数" data-language-autonym="中文" data-language-local-name="chinu" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – chinu lliterariu" lang="lzh" hreflang="lzh" data-title="實數" data-language-autonym="文言" data-language-local-name="chinu lliterariu" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%CC%8Dt-s%C3%B2%CD%98" title="Si̍t-sò͘ – chinu min nan" lang="nan" hreflang="nan" data-title="Si̍t-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="chinu min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – cantonés" lang="yue" hreflang="yue" data-title="實數" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12916#sitelinks-wikipedia" title="Editar los enllaces d'interllingua" class="wbc-editpage">Editar los enllaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nome"> <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/N%C3%BAmberu_real" title="Ver la páxina de conteníu [c]" accesskey="c"><span>Páxina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Alderique:N%C3%BAmberu_real&action=edit&redlink=1" rel="discussion" class="new" title="Alderique tocante al conteníu de la páxina (la páxina nun esiste) [t]" accesskey="t"><span>Alderique</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar variante de idioma" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">asturianu</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <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/N%C3%BAmberu_real"><span>Lleer</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&action=history" title="Versiones antigües d'esta páxina [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <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="Ferramientes" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramientes</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramientes</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">despintar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Aiciones </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/N%C3%BAmberu_real"><span>Lleer</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Xeneral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/N%C3%BAmberu_real" title="Llista de toles páxines wiki qu'enllacien equí [j]" accesskey="j"><span>Lo qu'enllaza equí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/N%C3%BAmberu_real" rel="nofollow" title="Cambios recientes nes páxines enllazaes dende esta [k]" accesskey="k"><span>Cambios rellacionaos</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=ast" title="Xubir ficheros [u]" accesskey="u"><span>Xubir ficheru</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Llista de toles páxines especiales [q]" accesskey="q"><span>Páxines especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&oldid=4178560" title="Enllaz permanente a esta revisión de la páxina"><span>Enllaz permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&action=info" title="Más información sobro esta páxina"><span>Información de la páxina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=N%C3%BAmberu_real&id=4178560&wpFormIdentifier=titleform" title="Información tocante a cómo citar esta páxina"><span>Citar esta páxina</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FN%25C3%25BAmberu_real"><span>Llograr la URL encurtiada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FN%25C3%25BAmberu_real"><span>Xenerar códigu QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprentar/esportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=N%C3%BAmberu+real"><span>Crear un llibru</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=N%C3%BAmberu_real&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=N%C3%BAmberu_real&printable=yes" title="Versión imprentable d'esta páxina [p]" accesskey="p"><span>Versión pa imprentar</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> N'otros proyeutos </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:Real_numbers" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12916" title="Enllaz al elementu del depósitu de datos coneutáu [g]" accesskey="g"><span>Elementu de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <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">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">despintar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-tradubot" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_real" title="Esti artículu foi traducíu automáticamente y precisa revisase manualmente"><img alt="Esti artículu foi traducíu automáticamente y precisa revisase manualmente" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/16px-Robot_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/24px-Robot_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Robot_icon.svg/32px-Robot_icon.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De Wikipedia</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="ast" dir="ltr"><table class="infobox plantia-xenerica" style="font-size:90%;width:25em"><tbody><tr><th colspan="2" style="text-align:center;font-size:125%;font-weight:bold;background-color: #acacac">Númberu real</th></tr><tr><td colspan="2" style="text-align:center;background-color: #cdcdcd"> tipu de númberu</td></tr><tr><td colspan="2" style="text-align:center;background-color: #cdcdcd"> númberu complexu y númberu</td></tr><tr><td colspan="2" style="text-align:center"> <span typeof="mw:File"><a href="/wiki/Ficheru:Number-line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/260px-Number-line.svg.png" decoding="async" width="260" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/390px-Number-line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/520px-Number-line.svg.png 2x" data-file-width="750" data-file-height="50" /></a></span></td></tr><tr><td colspan="2" style="text-align:right"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q12916" title="Cambiar los datos en Wikidata"><img alt="Cambiar los datos en Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/12px-Arbcom_ru_editing.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/18px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/24px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:N%C3%BAmeros_reales.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/220px-N%C3%BAmeros_reales.svg.png" decoding="async" width="220" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/330px-N%C3%BAmeros_reales.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/N%C3%BAmeros_reales.svg/440px-N%C3%BAmeros_reales.svg.png 2x" data-file-width="280" data-file-height="310" /></a><figcaption>Distintes clases de númberos reales.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheru:Real_number_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/300px-Real_number_line.svg.png" decoding="async" width="300" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/450px-Real_number_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/600px-Real_number_line.svg.png 2x" data-file-width="689" data-file-height="225" /></a><figcaption><a href="/w/index.php?title=Recta_real&action=edit&redlink=1" class="new" title="Recta real (la páxina nun esiste)">Recta real</a>.</figcaption></figure> <p>En <a href="/wiki/Matem%C3%A1tiques" title="Matemátiques">matemátiques</a>, el conxuntu de los <b>númberos reales</b> (denotado por <span class="Unicode">ℝ</span>) inclúi tanto a los <a href="/wiki/N%C3%BAmberu_racional" title="Númberu racional">númberos racionales</a> (positivos, negativos y el <a href="/wiki/Cero" title="Cero">cero</a>) como a los <a href="/wiki/N%C3%BAmberu_irracional" title="Númberu irracional">númberos irracionales</a>;<sup id="cite_ref-in_1-0" class="reference"><a href="#cite_note-in-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> y n'otru enfoque, <a href="/w/index.php?title=N%C3%BAmberu_trascendente&action=edit&redlink=1" class="new" title="Númberu trascendente (la páxina nun esiste)">trascendentes</a> y <a href="/w/index.php?title=N%C3%BAmberu_alxebraicu&action=edit&redlink=1" class="new" title="Númberu alxebraicu (la páxina nun esiste)">alxebraicos</a>. Los irracionales y los trascendentes<sup id="cite_ref-Tsipkin_2-0" class="reference"><a href="#cite_note-Tsipkin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (1970) non pueden espresase por aciu una <a href="/wiki/Fracci%C3%B3n" class="mw-redirect" title="Fracción">fracción</a> de dos enteros con denominador non nulu; tienen infinites cifres decimales aperiódicas, tales como: √<span style="border-top:1px solid #000">5</span>, <span class="texhtml">π</span>, el númberu real <span class="texhtml">log</span>2, que la so trescendencia foi enunciada por <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> nel sieglu XVIII.<sup id="cite_ref-Tsipkin_2-1" class="reference"><a href="#cite_note-Tsipkin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Los númberos reales pueden ser descritos y construyíos de delles formes, delles simples anque carentes del rigor necesario pa los propósitos formales de matemátiques y otres más complexes pero col rigor necesario pal trabayu matemáticu formal. </p><p>Mientres los sieglos XVI y XVII el <a href="/w/index.php?title=C%C3%A1lculu_infinitesimal&action=edit&redlink=1" class="new" title="Cálculu infinitesimal (la páxina nun esiste)">cálculu</a> avanzó enforma anque escarecía d'una base rigorosa, yá que nel momentu prescindíen del rigor y fundamentu lóxicu, tan esixente nos enfoques teóricos de l'actualidá, y usábense espresiones como «pequeñu», «llende», «averar ensin una definición precisa. Esto llevó a una serie de paradoxes y problemes lóxicos que fixeron evidente la necesidá de crear una base rigorosa pa la matemática, que consistió de <a href="/w/index.php?title=Definici%C3%B3n_(matem%C3%A1tica)&action=edit&redlink=1" class="new" title="Definición (matemática) (la páxina nun esiste)">definiciones</a> formales y rigoroses (anque verdaderamente téuniques) del conceutu de númberu real.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Nuna seición posterior van describise dos de les definiciones precises más avezaes anguaño: <a href="/w/index.php?title=Clases_d%27equivalencia&action=edit&redlink=1" class="new" title="Clases d'equivalencia (la páxina nun esiste)">clases d'equivalencia</a> de <a href="/w/index.php?title=Socesi%C3%B3n_de_Cauchy&action=edit&redlink=1" class="new" title="Socesión de Cauchy (la páxina nun esiste)">socesiones de Cauchy</a> de númberos racionales y <a href="/w/index.php?title=Cortadures_de_Dedekind&action=edit&redlink=1" class="new" title="Cortadures de Dedekind (la páxina nun esiste)">cortadures de Dedekind</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=1" title="Editar seición: Historia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=1" title="Editar el código fuente de la sección: Historia"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:Eye_of_Horus_(fractions).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Eye_of_Horus_%28fractions%29.svg/220px-Eye_of_Horus_%28fractions%29.svg.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Eye_of_Horus_%28fractions%29.svg/330px-Eye_of_Horus_%28fractions%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Eye_of_Horus_%28fractions%29.svg/440px-Eye_of_Horus_%28fractions%29.svg.png 2x" data-file-width="1279" data-file-height="723" /></a><figcaption></figcaption></figure> <p>Los <a href="/wiki/Antiguu_Exiptu" class="mw-redirect" title="Antiguu Exiptu">exipcios</a> dieron orixe per primer vegada a les <a href="/w/index.php?title=Fracci%C3%B3n_exipcia&action=edit&redlink=1" class="new" title="Fracción exipcia (la páxina nun esiste)">fracciones comunes</a> alredor del añu <a href="/wiki/1000_e.C." class="mw-redirect" title="1000 e.C.">1000 e. C.</a>; alredor del <a href="/wiki/500_e.C." title="500 e.C.">500 e. C.</a> un grupu de matemáticos <a href="/wiki/Antigua_Grecia" title="Antigua Grecia">griegos</a> lideraos por <a href="/wiki/Pit%C3%A1gores" title="Pitágores">Pitágores</a> diose cuenta de la necesidá de los <a href="/wiki/N%C3%BAmberu_irracional" title="Númberu irracional">númberos irracionales</a>. Los <a href="/wiki/N%C3%BAmberu_negativu" title="Númberu negativu">númberos negativos</a> fueron escurríos por matemáticos <a href="/wiki/India" title="India">indios</a> cerca del <a href="/wiki/600" title="600">600</a>, posiblemente reinventaos en <a href="/w/index.php?title=China_(rex%C3%B3n)&action=edit&redlink=1" class="new" title="China (rexón) (la páxina nun esiste)">China</a> pocu dempués, pero nun s'utilizaron n'<a href="/wiki/Europa" title="Europa">Europa</a> hasta'l <a href="/wiki/Sieglu_XVII" title="Sieglu XVII">sieglu XVII</a>, magar a finales del <a href="/wiki/Sieglu_XVIII" title="Sieglu XVIII">XVIII</a> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> refugó les soluciones negatives de les ecuaciones porque les consideraba irreales. Nesi sieglu, nel <a href="/wiki/Anal%C3%ADs_matem%C3%A1ticu" title="Analís matemáticu">cálculu</a> utilizábense númberos reales ensin una definición precisa, cosa que finalmente asocedió cola definición rigorosa fecha por <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> en <a href="/wiki/1871" title="1871">1871</a>. </p><p>En realidá, l'estudiu rigorosu de la construcción total de los númberos reales esixe tener amplios antecedentes de <a href="/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos">teoría de conxuntos</a> y <a href="/wiki/L%C3%B3xica_matem%C3%A1tica" title="Lóxica matemática">lóxica matemática</a>. Foi llograda la construcción y sistematización de los númberos reales nel sieglu XIX por dos grandes matemáticos europeos utilizando víes distintes: la teoría de conxuntos de Georg Cantor (encajamientos socesivos, cardinales finitos ya infinitos), per un sitiu, y l'analís matemáticu de <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> (vecindaes, redolaes y <a href="/w/index.php?title=Cortadures_de_Dedekind&action=edit&redlink=1" class="new" title="Cortadures de Dedekind (la páxina nun esiste)">cortadures de Dedekind</a>). Dambos matemáticos llograron la sistematización de los númberos reales na hestoria, non de manera bonal, sinón utilizando toles meyores previes na materia: dende l'antigua Grecia y pasando por matemáticos como <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a>, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>, <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a>, <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>, <a href="/wiki/Joseph-Louis_de_Lagrange" class="mw-redirect" title="Joseph-Louis de Lagrange">Lagrange</a>, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a>, <a href="/w/index.php?title=Cauchy&action=edit&redlink=1" class="new" title="Cauchy (la páxina nun esiste)">Cauchy</a> y <a href="/w/index.php?title=Weierstrass&action=edit&redlink=1" class="new" title="Weierstrass (la páxina nun esiste)">Weierstrass</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Evolución_del_conceutu_de_númberu"><span id="Evoluci.C3.B3n_del_conceutu_de_n.C3.BAmberu"></span>Evolución del conceutu de númberu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=2" title="Editar seición: Evolución del conceutu de númberu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=2" title="Editar el código fuente de la sección: Evolución del conceutu de númberu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sábese que los <a href="/w/index.php?title=Matem%C3%A1tiques_nel_Antiguu_Exiptu&action=edit&redlink=1" class="new" title="Matemátiques nel Antiguu Exiptu (la páxina nun esiste)">exipcios</a> y <a href="/w/index.php?title=Matem%C3%A1tica_babil%C3%B3nica&action=edit&redlink=1" class="new" title="Matemática babilónica (la páxina nun esiste)">babilónicos</a> faíen usu de fracciones (númberos racionales) na resolución de problemes práuticos. Sicasí, foi col desenvolvimientu de la matemática griega cuando se consideró l'aspeutu filosóficu de númberu. Los pitagóricos afayaron que les rellaciones harmóniques ente les notes musicales correspondíen a cocientes de númberos enteros, lo que los inspiró a buscar proporciones numbériques en toles demás coses, y espresar cola máxima «<i>tou ye númberu</i>». </p><p>Na matemática griega, dos magnitud son <i>conmensurables</i> si ye posible atopar una tercera tal que les primeres dos sían múltiplos de la postrera, esto ye, ye posible atopar una <i>unidá</i> común pa la que los dos magnitúes tengan una midida entera. El principiu pitagóricu de que tou númberu ye un cociente d'enteros, espresaba nesta forma que cualesquier dos magnitúes tienen de ser conmensurables. </p><p>Sicasí, l'ambiciosu proyeutu pitagóricu taramellóse ante'l problema de midir la diagonal d'un cuadráu, o la hipotenusa d'un triángulu rectángulu, pos nun ye conmensurable respectu de los catetos. En notación moderna, un triángulu rectángulu que los sos catetos miden 1, tien una hipotenusa que mide <a href="/w/index.php?title=Raiga%C3%B1u_cuadr%C3%A1u_de_dos&action=edit&redlink=1" class="new" title="Raigañu cuadráu de dos (la páxina nun esiste)">raigañu cuadráu de dos</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>: </p> <style data-mw-deduplicate="TemplateStyles:r4217957">.mw-parser-output .flexquote{display:flex;flex-direction:column;background-color:var(--background-color-neutral-subtle);border-left:3px solid var(--border-color-base);font-size:90%;margin:1em 4em;padding:.4em .8em}.mw-parser-output .flexquote>.flex{display:flex;flex-direction:row}.mw-parser-output .flexquote>.flex>.quote{width:100%}.mw-parser-output .flexquote>.flex>.separator{border-left:1px solid var(--border-color-base);border-top:1px solid var(--border-color-base);margin:.4em .8em}.mw-parser-output .flexquote>.cite{text-align:right}@media all and (max-width:600px){.mw-parser-output .flexquote>.flex{flex-direction:column}}</style><blockquote class="flexquote"> <div class="flex"> <div class="quote">Si por hipótesis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}={\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}={\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e7be04f0750138ae80b3822554a69f9f32c925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.202ex; height:5.343ex;" alt="{\displaystyle {\sqrt {2}}={\frac {p}{q}}}"></span> ye un númberu racional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9903bc1de26879e5fc4c7f78b54b952bcbb800f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}"></span> y ta amenorgáu, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2={\frac {p^{2}}{q^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2={\frac {p^{2}}{q^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c475d692e2822f2180219a815aca7faea86d73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.321ex; height:6.343ex;" alt="{\displaystyle 2={\frac {p^{2}}{q^{2}}}}"></span> d'onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2q^{2}=p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q^{2}=p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9d3cd7669a54a0b759112dbacadaa7816bea4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.618ex; height:3.009ex;" alt="{\displaystyle 2q^{2}=p^{2}}"></span>. <p>Si supónse que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> tienen un dos na so descomposición entós taría al cuadráu y por tanto sería una cantidá par nun llau de la igualdá cuando al otru llau ye impar. </p> Poro, el camientu que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> ye un númberu racional ten de ser falsa.</div> </div> </blockquote> <p>Surdió entós un dilema, yá que d'alcuerdu de primeres pitagóricu: tou númberu yera racional, mas la hipotenusa d'un triángulu rectángulu isósceles nun yera conmensurable colos catetos, lo cual implicó que d'equí p'arriba les magnitúes xeométriques y les cantidaes numbériques tendríen que tratase por separáu, fechu que tuvo consecuencies nel desenvolvimientu de la matemática mientres los dos milenios siguientes.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Los griegos desenvolvieron una xeometría basada en comparances (proporciones) de segmentos ensin faer referencia a valores numbéricos, usando diverses teoríes pa remanar el casu de midíes inconmensurables, como la <a href="/w/index.php?title=Teor%C3%ADa_de_proporciones_de_Eudoxo&action=edit&redlink=1" class="new" title="Teoría de proporciones de Eudoxo (la páxina nun esiste)">teoría de proporciones de Eudoxo</a>. Asina, los númberos irracionales permanecieron a partir d'entós escluyíos de l'aritmética yá que namái podíen ser trataos por aciu el métodu d'infinitos aproximamientos. Por casu, los pitagóricos atoparon (en notación moderna) que si <span class="texhtml"><a href="/w/index.php?title=Plant%C3%ADa:Fracci%C3%B3n&action=edit&redlink=1" class="new" title="Plantía:Fracción (la páxina nun esiste)">Plantía:Fracción</a></span> ye un aproximamientu a √<span style="border-top:1px solid #000">2</span> entós <span class="texhtml"><i>p</i></span> = <span class="texhtml"><i>a</i></span> + 2<span class="texhtml"><i>b</i></span> y <span class="texhtml"><i>q</i></span> = <span class="texhtml"><i>a</i></span> + <span class="texhtml"><i>b</i></span> son tales que <span class="texhtml"><a href="/w/index.php?title=Plant%C3%ADa:Fracci%C3%B3n&action=edit&redlink=1" class="new" title="Plantía:Fracción (la páxina nun esiste)">Plantía:Fracción</a></span> ye un aproximamientu más precisu. Repitiendo'l procesu nuevamente llógrense mayores númberos que dan un meyor aproximamientu.<sup id="cite_ref-Stillwell_5-0" class="reference"><a href="#cite_note-Stillwell-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Yá que los llargores qu'espresen los númberos irracionales podíen ser llograes por aciu procesos xeométricos senciellos pero, aritméticamente, namái por aciu procesos d'infinitos aproximamientos, anició que mientres 2000 años la teoría de los númberos reales fora esencialmente xeométrica, identificando los númberos reales colos puntos d'una llinia recta. </p><p>Nueves meyores nel conceutu de númberu real esperaron hasta los sieglos XVI y XVII, col desenvolvimientu de la notación alxebraica, lo que dexó la manipulación y operación de cantidaes ensin faer referencia a segmentos y llargores. Por casu, atopáronse fórmules pa resolver ecuaciones de segundu y tercer grau de forma mecánica por aciu <a href="/wiki/Algoritmu" title="Algoritmu">algoritmos</a>, que incluyíen raigaños ya inclusive, n'ocasiones, «númberos non reales» (lo qu'agora conocemos como <a href="/wiki/N%C3%BAmberu_complexu" title="Númberu complexu">númberos complexos</a>). Sicasí, nun esistía entá un conceutu formal de númberu e siguíase dando primacía a la <a href="/wiki/Xeometr%C3%ADa" title="Xeometría">xeometría</a> como fundamentu de tola matemática. Inclusive col desenvolvimientu de la <a href="/wiki/Xeometr%C3%ADa_anal%C3%ADtica" title="Xeometría analítica">xeometría analítica</a> esti puntu de vista calteníase vixente, pos <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a> refugaba la idea que la xeometría pudiera encontase en númberos, yá que para él la nueva área yera a cencielles una ferramienta pa resolver problemes xeométricos. </p><p>Darréu, la invención del <a href="/w/index.php?title=C%C3%A1lculu_infinitesimal&action=edit&redlink=1" class="new" title="Cálculu infinitesimal (la páxina nun esiste)">cálculu</a> abrió un periodu de grandes meyores matemátiques, con nuevos y poderosos métodos que dexaron per vegada primera atacar los problemes rellacionaos colo infinito por aciu el conceutu de <a href="/w/index.php?title=Llende_d%27una_funci%C3%B3n_llende&action=edit&redlink=1" class="new" title="Llende d'una función llende (la páxina nun esiste)">Llende d'una función llende</a>. Asina, un númberu irracional pudo ser entendíu como la llende d'una suma infinita de númberos racionales (por casu, la so espansión decimal). Como amuesa, el númberu <span class="texhtml">π</span> puede estudiase de forma alxebraica (ensin apelar a la intuición xeométrica) por aciu la serie: </p> <style data-mw-deduplicate="TemplateStyles:r4219090">.mw-parser-output .ecuacion{padding:5px 10px;background-color:var(--background-color-base);color:var(--color-base);margin-left:30px;margin-bottom:0.8em;margin-top:0.5em;min-width:50%}.mw-parser-output .ecuacion .referencia{float:right;width:10%;text-align:end}.mw-parser-output .ecuacion cite{font-style:normal}</style><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)=4\sum _{k=0}^{\infty }(-1)^{k}{\frac {1}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)=4\sum _{k=0}^{\infty }(-1)^{k}{\frac {1}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a95bedd6231dc962f61fef751efa9d0b8b306f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.729ex; height:7.009ex;" alt="{\displaystyle \pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)=4\sum _{k=0}^{\infty }(-1)^{k}{\frac {1}{2k+1}}}"></span> </p> </blockquote> <p>ente munches otres espresiones similares. Aquel día, el conceutu intuitivu de númberu real yera yá'l modernu, identificando ensin problema un segmentu cola midida del so llargor (racional o non). El cálculu abrió'l pasu al <a href="/wiki/Anal%C3%ADs_matem%C3%A1ticu" title="Analís matemáticu">analís matemáticu</a>, qu'estudia conceutos como continuidá, converxencia, etc. Pero l'analís nun cuntaba con definiciones rigoroses y munches de les demostraciones apelaben entá a la intuición xeométrica. Esto traxo a una serie de paradoxes ya imprecisiones. </p> <div class="mw-heading mw-heading2"><h2 id="Notación"><span id="Notaci.C3.B3n"></span>Notación</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=3" title="Editar seición: Notación" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=3" title="Editar el código fuente de la sección: Notación"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheru:Latex_real_numbers_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/150px-Latex_real_numbers_square.svg.png" decoding="async" width="150" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/225px-Latex_real_numbers_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/300px-Latex_real_numbers_square.svg.png 2x" data-file-width="343" data-file-height="341" /></a><figcaption></figcaption></figure> <p>Los númberos reales espresar con decimales que tienen una secuencia infinita de díxitos a la derecha de la coma decimal, como por casu 324,8232. Frecuentemente tamién se sub representen con tres puntos consecutivos a la fin (324,823211247…), lo que significaría qu'entá falten más díxitos decimales, pero que se consideren ensin importancia. </p><p>Les midíes nes <a href="/wiki/F%C3%ADsica" title="Física">ciencies físiques</a> son siempres un aproximamientu a un númberu real. Non yá ye más concisu escribilos con forma de fracción decimal (esto ye, <a href="/wiki/N%C3%BAmberu_racional" title="Númberu racional">númberos racionales</a> que pueden ser escritos como proporciones, con un denominador exactu) sinón que, sía que non, suple íntegramente el conceutu y significáu del númberu real. Nel analís matemáticu los númberos reales son oxetu principal d'estudiu. Puede dicise que los númberos reales son la ferramienta de trabayu de les matemátiques de la continuidá, como'l cálculu y l'analís matemáticu, ente que los númberos enteros ser de les <a href="/wiki/Matem%C3%A1tiques_discretes" title="Matemátiques discretes">matemátiques discretes</a>, nes que ta ausente la continuidá. </p><p>Dizse qu'un númberu real ye <b>recursivo</b> si los sos díxitos pueden espresase por un algoritmu recursivo. Un númberu <b>non-recursivo</b> ye aquél que ye imposible d'especificar explícitamente. Aun así, la escuela rusa de <a href="/w/index.php?title=Constructivismo_(matem%C3%A1tiques)&action=edit&redlink=1" class="new" title="Constructivismo (matemátiques) (la páxina nun esiste)">constructivismo</a> supón que tolos númberos reales son recursivos. </p><p>Los ordenadores namái pueden averase a los númberos reales por númberos racionales; de toes formes, dellos programes d'ordenador pueden tratar un númberu real de manera exacta usando la so definición alxebraica (por casu, "<span style="vertical-align:text-top; line-height:100%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span></span>") en cuenta de el so respeutivu aproximamientu decimal. </p><p>Los matemáticos usen el símbolu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> (o, d'otra forma, <span style="vertical-align:text-top; line-height:100%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></span></span>, la lletra "<a href="/wiki/R" title="R">R</a>" en negrina) pa representar el conxuntu de tolos númberos reales. La <a href="/w/index.php?title=Tabla_de_s%C3%ADmbolos_matem%C3%A1ticos&action=edit&redlink=1" class="new" title="Tabla de símbolos matemáticos (la páxina nun esiste)">notación matemática</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> referir a un espaciu de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> dimensiones de los númberos reales; por casu, un valor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <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>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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} ^{3}}"></span> consiste de tres númberu reales y determina un llugar nun espaciu de tres dimensiones. </p><p>En matemática, la pallabra "real" úsase como axetivu, col significáu de que'l campu subxacente ye'l campu de los númberos reales. Por casu, <i><a href="/w/index.php?title=Matriz_(matem%C3%A1tica)&action=edit&redlink=1" class="new" title="Matriz (matemática) (la páxina nun esiste)">matriz</a> real</i>, <i><a href="/w/index.php?title=Funci%C3%B3n_real&action=edit&redlink=1" class="new" title="Función real (la páxina nun esiste)">función real</a></i>, y <i><a href="/w/index.php?title=%C3%81lxebra_de_Lie&action=edit&redlink=1" class="new" title="Álxebra de Lie (la páxina nun esiste)">Álxebra de Lie</a> real</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Tipos_de_númberos_reales"><span id="Tipos_de_n.C3.BAmberos_reales"></span>Tipos de númberos reales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=4" title="Editar seición: Tipos de númberos reales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=4" title="Editar el código fuente de la sección: Tipos de númberos reales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Racionales_ya_irracionales">Racionales ya irracionales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=5" title="Editar seición: Racionales ya irracionales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=5" title="Editar el código fuente de la sección: Racionales ya irracionales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un númberu real pue ser un <i><a href="/wiki/N%C3%BAmberu_racional" title="Númberu racional">númberu racional</a></i> o un <i><a href="/wiki/N%C3%BAmberu_irracional" title="Númberu irracional">númberu irracional</a></i>. Los númberos racionales son aquellos que pueden espresase como'l <a href="/wiki/Cociente_(aritm%C3%A9tica)" class="mw-redirect" title="Cociente (aritmética)">cociente</a> de dos númberos enteros, tal como 3/4, -21/3, 5, 0, 1/2, ente que los irracionales son tolos demás. Los númberos racionales tamién pueden describise como aquellos que la so representación decimal ye eventualmente periódica, ente que los irracionales tienen una espansión decimal aperiódica: </p> <dl><dt>Exemplos</dt> <dd>1/4 = 0,25<b>0</b>0<b>0</b>0... Ye un númberu racional yá que ye periódicu a partir del tercer númberu decimal .</dd> <dd>5/7 = 0,<b>714285</b>714285<b>714285</b>7.... Ye racional y tien un periodu de llargor 6 (repite 714285) .</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}=1{\text{,}}456465591386194\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> <mn>456465591386194</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}=1{\text{,}}456465591386194\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63134a4a2bf80accd9c7f5b6415fbd4e713708fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.392ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}=1{\text{,}}456465591386194\ldots }"></span> ye irracional y la so espansión decimal ye aperiódica .</dd></dl> <p>El conxuntu de los númberos racionales designar por aciu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> . </p> <div class="mw-heading mw-heading3"><h3 id="Alxebraicos_y_trescendentes">Alxebraicos y trescendentes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=6" title="Editar seición: Alxebraicos y trescendentes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=6" title="Editar el código fuente de la sección: Alxebraicos y trescendentes"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Otra forma de clasificar los númberos reales ye en <i><a href="/w/index.php?title=N%C3%BAmberu_alxebraicu&action=edit&redlink=1" class="new" title="Númberu alxebraicu (la páxina nun esiste)">alxebraicos</a></i> y <i><a href="/w/index.php?title=N%C3%BAmberu_trascendente&action=edit&redlink=1" class="new" title="Númberu trascendente (la páxina nun esiste)">trascendentes</a></i>. Un númberu ye alxebraicu si esiste un <a href="/wiki/Polinomiu" title="Polinomiu">polinomiu</a> de coeficientes racionales que lo tien por raigañu y ye trascendente en casu contrariu. Obviamente, tolos númberos racionales son alxebraicos: si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9903bc1de26879e5fc4c7f78b54b952bcbb800f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}"></span> ye un númberu racional, con <i>p</i> enteru y <i>q</i> natural, entós ye raigañu de la ecuación <i>qx</i>=<i>p</i>. Sicasí, non tolos númberos alxebraicos son racionales. </p> <dl><dt>Exemplos</dt> <dd>El númberu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4137154ff2c27d413f3fe21ed11e8a0c3633b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.937ex; height:5.843ex;" alt="{\displaystyle {\frac {{\sqrt[{3}]{7}}+1}{2}}}"></span> ye alxebraicu yá que ye un raigañu del polinomiu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x^{3}-6x^{2}+3x-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{3}-6x^{2}+3x-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c61f611926dc828318ef6eadc28296443e64db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.269ex; height:2.843ex;" alt="{\displaystyle 4x^{3}-6x^{2}+3x-4}"></span></dd> <dd>Un exemplu de númberu trascendente ye <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln 3=1{\text{,}}09861228866811\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mn>3</mn> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> <mn>09861228866811</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln 3=1{\text{,}}09861228866811\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d8aa40ed3e5245cc01a9c5add3c298b0ad8b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.781ex; height:2.509ex;" alt="{\displaystyle \ln 3=1{\text{,}}09861228866811\ldots }"></span></dd></dl> <p>El conxuntu de los númberos alxebraicos designar por aciu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Computables_y_irreductibles">Computables y irreductibles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=7" title="Editar seición: Computables y irreductibles" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=7" title="Editar el código fuente de la sección: Computables y irreductibles"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un númberu real dizse computable si tien una <a href="/w/index.php?title=Complexid%C3%A1_de_Kolmog%C3%B3rov&action=edit&redlink=1" class="new" title="Complexidá de Kolmogórov (la páxina nun esiste)">complexidá de Kolmogórov</a> finita, esto ye, si puede escribise un programa informáticu d'estensión finita que xenere los díxitos de dichu númberu. Si un númberu real nun ye computable dizse irreductible. Una definición de númberu irreductible ye: </p><p>El conxuntu de númberos reales computables designar por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{\rm {comp}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\rm {comp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1098ae281c1cbd6328da48f7dccf2ff1a0f045a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.745ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{\rm {comp}}}"></span>. Obviamente los racionales y los alxebraicos son númberos computables. De fechu tiense la siguiente inclusión: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \subset \mathbb {A} \subset \mathbb {R} _{\rm {comp}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \subset \mathbb {A} \subset \mathbb {R} _{\rm {comp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2fc088cfdbb2c3506aa87a79c76deb174d5ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.428ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \subset \mathbb {A} \subset \mathbb {R} _{\rm {comp}}}"></span> </p> </blockquote> <p>Amás tiense que toos estos conxuntos son <a href="/w/index.php?title=Conxuntu_numerable&action=edit&redlink=1" class="new" title="Conxuntu numerable (la páxina nun esiste)">numerables</a>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{card}}|\mathbb {Q} |={\text{card}}|\mathbb {A} |={\text{card}}|\mathbb {R} _{\rm {comp}}|=\aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>card</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>card</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>card</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{card}}|\mathbb {Q} |={\text{card}}|\mathbb {A} |={\text{card}}|\mathbb {R} _{\rm {comp}}|=\aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a40f89d6a067c89584998d6f77349c132640ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.079ex; height:3.009ex;" alt="{\displaystyle {\text{card}}|\mathbb {Q} |={\text{card}}|\mathbb {A} |={\text{card}}|\mathbb {R} _{\rm {comp}}|=\aleph _{0}}"></span> </p> </blockquote> <p>Esto implica que'l conxuntu de tolos númberos computables ye un conxuntu de <a href="/w/index.php?title=Teor%C3%ADa_de_la_midida_midida_nula&action=edit&redlink=1" class="new" title="Teoría de la midida midida nula (la páxina nun esiste)">teoría de la midida midida nula</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Construcciones_del_conxuntu_de_númberos_reales"><span id="Construcciones_del_conxuntu_de_n.C3.BAmberos_reales"></span>Construcciones del conxuntu de númberos reales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=8" title="Editar seición: Construcciones del conxuntu de númberos reales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=8" title="Editar el código fuente de la sección: Construcciones del conxuntu de númberos reales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Presentación_axomática"><span id="Presentaci.C3.B3n_axom.C3.A1tica"></span>Presentación axomática</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=9" title="Editar seición: Presentación axomática" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=9" title="Editar el código fuente de la sección: Presentación axomática"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Foi propuestu pol matemáticu alemán David Hilbert. En testos actuales de cálculu y analís matemáticu apaecen enunciaos equivalentes al de Hilbert.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r4219085">.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">Artículu principal: <a href="/w/index.php?title=Axomes_de_los_n%C3%BAmberos_reales&action=edit&redlink=1" class="new" title="Axomes de los númberos reales (la páxina nun esiste)">Axomes de los númberos reales</a></div> <p>Esisten distintes formes de construyir el conxuntu de los númberos reales a partir d'axomes, siendo la carauterización más común, el conocíu como <i>métodu direutu</i> qu'introduz el sistema (ℝ, +,., ≤), onde los elementos de llámense <i>númberos reales</i>, + y. son dos operaciones en ℝ, ≤ ye una rellación d'orde en ℝ.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Preséntase una variante axomática, por aciu les siguientes trés propiedaes: <style data-mw-deduplicate="TemplateStyles:r4219309">.mw-parser-output .teorema-contenedor{min-width:50%;max-width:77%}.mw-parser-output .teorema{padding:.5em 2em .5em 1.5em;padding-right:2em;padding-left:1.5em;padding-bottom:0.5em;padding-top:0.5em;border:1px solid var(--border-color-base,#49768C);font-family:Georgia,serif}.mw-parser-output .teorema-pie{margin-top:-1em;text-align:right}</style> </p> <table class="teorema-contenedor"> <tbody><tr> <td><blockquote class="teorema"> <p>Un conxuntu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,+,\cdot ,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,+,\cdot ,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87b324c4dbba246bb592066331864a77cc8d729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:2.843ex;" alt="{\displaystyle (K,+,\cdot ,\leq )}"></span> ye <i>el conxuntu de los númberos reales</i> si satisfai les siguientes trés condiciones: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,+,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d474a4df1822902bbe09dab84bb9872f0019d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.398ex; height:2.843ex;" alt="{\displaystyle (K,+,\cdot )}"></span> ye un <a href="/w/index.php?title=Campu_(matem%C3%A1tiques)&action=edit&redlink=1" class="new" title="Campu (matemátiques) (la páxina nun esiste)">campu</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b40c3e4804b6bb2c3d024aa0b8f233aa5c5d11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.843ex;" alt="{\displaystyle (K,\leq )}"></span> ye un <a href="/w/index.php?title=Orde_total&action=edit&redlink=1" class="new" title="Orde total (la páxina nun esiste)">conxuntu totalmente ordenáu</a> y l'orde ye compatible coles operaciones del campu: <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span> entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+c\leq b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>≤</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c\leq b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878af0e64fb866c2e3e1955d3bda821def919ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.02ex; height:2.343ex;" alt="{\displaystyle a+c\leq b+c}"></span>;</dd> <dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9667688ebbea39a0076edd3c87a28799c1055db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.268ex; height:2.343ex;" alt="{\displaystyle 0\leq c}"></span> entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac\leq bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>c</mi> <mo>≤</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ac\leq bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd77306a687391107efb2ab53af7cad02020d4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.34ex; height:2.343ex;" alt="{\displaystyle ac\leq bc}"></span>.</dd></dl></li> <li>El conxuntu <i>K</i> ye completu: satisfai'l <a href="/w/index.php?title=Axoma_del_supremu&action=edit&redlink=1" class="new" title="Axoma del supremu (la páxina nun esiste)">axoma del supremu</a>: <dl><dd>Tou conxuntu non vacíu y acutáu superiormente tien un <a href="/w/index.php?title=Supremu_(matem%C3%A1tiques)&action=edit&redlink=1" class="new" title="Supremu (matemátiques) (la páxina nun esiste)">supremu</a>.</dd></dl></li></ol> </blockquote> </td></tr></tbody></table> <ul><li>L'axoma del supremu ye una variante del <i>Principiu de Weirstrass" que diz que toa socesión de númberos reales acutada superiormente tien supremu Les</i></li></ul> <p>primeres dos condiciones definen el conceutu de <i>campu ordenáu</i>, ente que la tercer propiedá ye de naturaleza topolóxica y ye la qu'estrema al conxuntu de los númberos reales de tolos demás campos ordenaos. Hai que faer notar que, en principiu pueden esistir distintos conxuntos que satisfaigan les mesmes condiciones y que podríen ser distintes al conxuntu de los númberos reales, pero un teorema establez que si eso asocediera, dambes estructures seríen esencialmente la mesma. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219309"> <table class="teorema-contenedor"> <tbody><tr> <td><blockquote class="teorema"> <p>Cualquier campu ordenáu que cumpla los trés propiedaes mentaes ye <a href="/w/index.php?title=Isomorfu&action=edit&redlink=1" class="new" title="Isomorfu (la páxina nun esiste)">isomorfu</a> al conxuntu de los númberos reales. </p> </blockquote> </td></tr></tbody></table> <p>En vista de lo anterior podemos falar de <i>el</i> conxuntu de los númberos reales (y non de <i>un</i> conxuntu de númberos reales) y estableciendo la so unicidá puede usase el símbolu ℝ pa representalo. </p><p>Al enunciar la tercer propiedá n'ocasiones especifícase que ℝ ye completu nel sentíu de Dedekind, pos esisten otros axomes que pueden usase y que, asumiendo les primeres dos condiciones, toos son lóxicamente equivalentes. Dalgunos d'estos son: </p> <ul><li>(Cauchy) El conxuntu <i>K</i> cumple que cualesquier <a href="/w/index.php?title=Socesi%C3%B3n_de_Cauchy&action=edit&redlink=1" class="new" title="Socesión de Cauchy (la páxina nun esiste)">socesión de Cauchy</a> ye converxente.</li> <li>(Bolzano-Weierstrass) El conxuntu <i>K</i> cumple que cualquier socesión acutada tien una subsocesión converxente.</li> <li>Cualquier socesión decreciente d'intervalos zarraos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⊇</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780fd7e9b7632173d223b2650a6788c644a1ed3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.251ex; height:2.509ex;" alt="{\displaystyle I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdots }"></span> tien interseición non vacida.</li></ul> <p>Caúna de les primeres dos propiedaes mentaes al entamu de la seición correspuenden de la mesma a otra serie d'axomes, de cuenta que si fai un desglose, puede caracterizase'l conxuntu de los númberos reales como un conxuntu que satisfaiga la siguiente llista d'axomes. </p> <ol><li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8234bfe0026070a0777d6bdd0bc06cc27723c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.844ex; height:2.509ex;" alt="{\displaystyle x+y\in \mathbb {R} }"></span> (Pesllera na suma)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=y+x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=y+x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c22a849862f6ca7d7a9413a58431a37f1e4e0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.137ex; height:2.343ex;" alt="{\displaystyle x+y=y+x\,}"></span> (Conmutatividad na suma)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)+z=x+(y+z)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)+z=x+(y+z)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30ebdda2a6519f0951fd1443021537da01e2b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.612ex; height:2.843ex;" alt="{\displaystyle (x+y)+z=x+(y+z)\,}"></span> (Asociatividad na suma)</li> <li>Esisti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b744aa80fd21e49f206ec213a0889eb81b80ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:2.176ex;" alt="{\displaystyle 0\in \mathbb {R} }"></span> de manera que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+0=x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+0=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb55f2baaf8b0239f3eb18c4a65a1f6857744cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.148ex; height:2.343ex;" alt="{\displaystyle x+0=x\,}"></span> pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span> (Neutru aditivu)</li> <li>Pa cada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span> esiste un elementu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b56dc64b4ff74b700100bd6824b9f8768b61a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.657ex; height:2.343ex;" alt="{\displaystyle -x\in \mathbb {R} }"></span> tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x+x=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x+x=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed80cf34044d6ab25ac252efd06a35edcf1a0ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.956ex; height:2.343ex;" alt="{\displaystyle -x+x=0\,}"></span> (Inversu aditivu)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e333aa1d93c49538c51c89f727870dcdd1b51e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.004ex; height:2.509ex;" alt="{\displaystyle xy\in \mathbb {R} }"></span> (Pesllera na multiplicación)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=yx\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe6a78cafda93cef2011f48efdc2737fc411e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.456ex; height:2.009ex;" alt="{\displaystyle xy=yx\,}"></span> (Conmutatividad na multiplicación)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)z=x(yz)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5d15bebb37a57dd939517d4ae564f22c115e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.251ex; height:2.843ex;" alt="{\displaystyle (xy)z=x(yz)\,}"></span> (Asociatividad na multiplicación)</li> <li>Esisti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71daf1dce3cd33e183369ba029eadafc8806294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:2.176ex;" alt="{\displaystyle 1\in \mathbb {R} }"></span> de manera que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot {1}=1\cdot {x}=x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot {1}=1\cdot {x}=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf54a8118939cecb356b25aae64f511433d7b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.256ex; height:2.176ex;" alt="{\displaystyle x\cdot {1}=1\cdot {x}=x\,}"></span> pa cualesquier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span> (Neutru multiplicativu)</li> <li>Pa cada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0,x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0,x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420cbf2b276bdbeb06f2e4a76b54e54e96f6e360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.473ex; height:2.676ex;" alt="{\displaystyle x\neq 0,x\in \mathbb {R} }"></span> esiste un elementu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e1c2bb31fb715adc690dc2b6e5fc8b86c3f3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.181ex; height:2.676ex;" alt="{\displaystyle x^{-1}\in \mathbb {R} }"></span> tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}x=1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}x=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9042e5465df319cb45044801ce60aeefb2c9c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.64ex; height:2.676ex;" alt="{\displaystyle x^{-1}x=1\,}"></span> (Inversu multiplicativu)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(y+z)=xy+xz\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(y+z)=xy+xz\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b84653f14ad992f76e670f8c64da779b7077a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.452ex; height:2.843ex;" alt="{\displaystyle x(y+z)=xy+xz\,}"></span> (Distributividad de la multiplicación na suma)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>, entós cumplir namái una d'estes: (<a href="/w/index.php?title=Llei_de_tricotom%C3%ADa&action=edit&redlink=1" class="new" title="Llei de tricotomía (la páxina nun esiste)">Tricotomía</a>) <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ce4ac40139ac6343aec4897ccfc9606dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x<y\,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee0731e088428f41ec0dde3374f87a8874360ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle y<x\,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8def71aa31e67583ef8e0eda1392b3dbd596dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.009ex;" alt="{\displaystyle x=y\,}"></span></li></ul></li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ce4ac40139ac6343aec4897ccfc9606dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x<y\,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<z\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53010d5b6e3f21c32595580d429a3265fadbfbad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.729ex; height:2.176ex;" alt="{\displaystyle y<z\,}"></span> entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<z\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9687ff8fbac93bc671e7fbcad52cb7aa1f41fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.903ex; height:1.843ex;" alt="{\displaystyle x<z\,}"></span> (Transitividá)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ce4ac40139ac6343aec4897ccfc9606dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x<y\,}"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+z<y+z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo><</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+z<y+z\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aef9934183ec2938bd7636a5611f0439382c9a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.828ex; height:2.343ex;" alt="{\displaystyle x+z<y+z\,}"></span> (Monotonía na suma)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34e73e57f1e73a936ce1d3ccffed2c86cb2ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.16ex; height:2.509ex;" alt="{\displaystyle x,y,z\in \mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ce4ac40139ac6343aec4897ccfc9606dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x<y\,}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<z\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636814518eb6ba5f539c725283efdfab533d0883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.736ex; height:2.176ex;" alt="{\displaystyle 0<z\,}"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xz<yz\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>z</mi> <mo><</mo> <mi>y</mi> <mi>z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xz<yz\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8524def366dde53fc6dc6bb003c6a4975533c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.147ex; height:2.176ex;" alt="{\displaystyle xz<yz\,}"></span> (Monotonía na multiplicación)</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198307a9ef3ba7869c232066445b5dd23fc256ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.55ex; height:2.176ex;" alt="{\displaystyle Y\subset \mathbb {R} }"></span> ye un conxuntu non vacíu acutáu superiormente en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, entós <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e3deb85bd2bfe306da34e635f7bfb2926daf8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.16ex; height:2.009ex;" alt="{\displaystyle Y\,}"></span> tien supremu en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> (<a href="/w/index.php?title=Axoma_del_supremu&action=edit&redlink=1" class="new" title="Axoma del supremu (la páxina nun esiste)">Axoma del supremu</a>)</li></ol> <p>Los axomes del 1 al 15 correspuenden a la estructura más xeneral de cuerpu ordenáu. L'últimu axoma ye'l qu'estrema <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> d'otros cuerpos ordenaos como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>. Tien De señalase que los axomes 1 a 15 nun constitúin una <a href="/w/index.php?title=Teor%C3%ADa_(l%C3%B3xica)&action=edit&redlink=1" class="new" title="Teoría (lóxica) (la páxina nun esiste)">teoría categórica</a> yá que puede demostrase qu'almiten siquier un <a href="/w/index.php?title=Teor%C3%ADa_de_modelos&action=edit&redlink=1" class="new" title="Teoría de modelos (la páxina nun esiste)">modelu</a> <a href="/w/index.php?title=Analises_non-est%C3%A1ndar&action=edit&redlink=1" class="new" title="Analises non-estándar (la páxina nun esiste)">non estándar</a> distintu de los númberos reales, que ye precisamente'l modelu nel que se basa la construcción de los <a href="/w/index.php?title=N%C3%BAmberu_hiperreal&action=edit&redlink=1" class="new" title="Númberu hiperreal (la páxina nun esiste)">númberos hiperreales</a> </p> <div class="mw-heading mw-heading3"><h3 id="Construcción_por_númberos_decimales"><span id="Construcci.C3.B3n_por_n.C3.BAmberos_decimales"></span>Construcción por númberos decimales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=10" title="Editar seición: Construcción por númberos decimales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=10" title="Editar el código fuente de la sección: Construcción por númberos decimales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consideramos los <a href="/wiki/N%C3%BAmberu_decimal" class="mw-redirect" title="Númberu decimal">númberos decimales</a> como los conocemos intuitivamente. Sabemos que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =3,1415926535897932384626\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>1415926535897932384626</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3,1415926535897932384626\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093e58947a4b8ffa90a439b5f279ad29014843ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.311ex; height:2.509ex;" alt="{\displaystyle \pi =3,1415926535897932384626\dots }"></span>, esto ye, el <a href="/wiki/N%C3%BAmberu_pi" class="mw-redirect" title="Númberu pi">númberu π</a> esprésase como'l <a href="/wiki/N%C3%BAmberu_enteru" title="Númberu enteru">númberu enteru</a> 3 y una secuencia infinita de <i>díxitos</i> 1, 4, 1, 5, 9, 2, etc. </p><p>Un númberu decimal esprésase entós como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.d_{1}d_{2}d_{3}d_{4}\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.d_{1}d_{2}d_{3}d_{4}\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28b2bd0e455d68f63151321c795ae1dfdaa3215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.527ex; height:2.509ex;" alt="{\displaystyle x.d_{1}d_{2}d_{3}d_{4}\dots }"></span> onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ye un númberu enteru y cada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"></span> ye un elementu del conxuntu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,3,4,5,6,7,8,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,3,4,5,6,7,8,9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87e79c813bba1455d02d5f887975b42ac3b7adaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.255ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,3,4,5,6,7,8,9\}}"></span>. Amás, consideramos que nun esisten les <a href="/wiki/0,9_peri%C3%B3dicu" title="0,9 periódicu">coles de 9</a>. </p><p>Al conxuntu de tolos númberos decimales onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ye un númberu enteru positivu se -y denota por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{+}}"></span> y llámase-y el conxuntu de los númberos <i>reales positivos</i>. </p><p>Al conxuntu de tolos númberos decimales onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ye un númberu enteru negativu se -y denota por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158001a03e958f49f5885033776a420fc47b7267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{-}}"></span> y llámase-y el conxuntu de los númberos <i>reales negativos</i>. </p><p>Al númberu decimal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,00000\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>00000</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,00000\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66faffa5f63067db2c422588235b5a11eaebf50c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.119ex; height:2.509ex;" alt="{\displaystyle 0,00000\dots }"></span> llámase-y <i>cero</i>. </p><p>Al conxuntu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{+}\cup \mathbb {R} ^{-}\cup \{0,00000\dots \}}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>00000</mn> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}\cup \mathbb {R} ^{-}\cup \{0,00000\dots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270406d385016f1d69d51eee7f699cf7bc5b86ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.987ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{+}\cup \mathbb {R} ^{-}\cup \{0,00000\dots \}}"></span> se -y denota por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> y llámase-y conxuntu de <i>númberos reales</i>. </p><p>Defínese la rellación d'<a href="/w/index.php?title=Orde_total&action=edit&redlink=1" class="new" title="Orde total (la páxina nun esiste)">orde total</a> de los númberos decimales como # <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0>x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>></mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0>x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f439c437a360ce47ac647d4b11a08bc72bdcf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle 0>x\,}"></span> pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5bdf8e7b8800bb13f52e18866004d32931d252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {R} ^{-}}"></span> </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967157231724bdce0640c62d56928d306e3e7ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x>y\,}"></span> siempres que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f3ab5fc33241badf9bc8f08b9bee1a3616e357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {R} ^{+}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {R} ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {R} ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2866554448e23ded306add4a946a2fbe369302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.185ex; height:2.843ex;" alt="{\displaystyle y\in \mathbb {R} ^{-}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf3dbb12d9694caacbda897b8408618db0c4903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle x>0\,}"></span> pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f3ab5fc33241badf9bc8f08b9bee1a3616e357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {R} ^{+}}"></span></li> <li>Daos dos númberos reales cualesquier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a.a_{1}a_{2}a_{3}a_{4}\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>.</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a.a_{1}a_{2}a_{3}a_{4}\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512aec512e208bee240612dcc583265efd7157ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.938ex; height:2.009ex;" alt="{\displaystyle x=a.a_{1}a_{2}a_{3}a_{4}\dots }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b.b_{1}b_{2}b_{3}b_{4}\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b.b_{1}b_{2}b_{3}b_{4}\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e629bfb8fcda2d62effe7aad3a058ab970af35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.603ex; height:2.509ex;" alt="{\displaystyle y=b.b_{1}b_{2}b_{3}b_{4}\dots }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967157231724bdce0640c62d56928d306e3e7ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x>y\,}"></span> en cualesquier de los casos siguientes: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a472db05ed6a68e5fbd8ba96a86f6041e60fbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.713ex; height:2.176ex;" alt="{\displaystyle a>b\,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d431573107d7d12877cd7f1eaee6f0e99532e330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.713ex; height:2.176ex;" alt="{\displaystyle a=b\,}"></span> y amás esiste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=b_{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=b_{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca62166f7bdfbf1b5d33989447912cd8177299c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.312ex; height:2.509ex;" alt="{\displaystyle a_{i}=b_{i}\,}"></span> pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a125ffac87dde409b5799717bfcbe4121b91ad04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i<n}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}>b_{n}\,}"> <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> <mo>></mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}>b_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f96fae18cfb2405cabc11263b42497165e3764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.15ex; height:2.509ex;" alt="{\displaystyle a_{n}>b_{n}\,}"></span></li></ul></li></ol> <div class="mw-heading mw-heading3"><h3 id="Construcción_por_cortadures_de_Dedekind"><span id="Construcci.C3.B3n_por_cortadures_de_Dedekind"></span>Construcción por cortadures de Dedekind</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=11" title="Editar seición: Construcción por cortadures de Dedekind" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=11" title="Editar el código fuente de la sección: Construcción por cortadures de Dedekind"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=Cortadures_de_Dedekind&action=edit&redlink=1" class="new" title="Cortadures de Dedekind (la páxina nun esiste)">Cortadures de Dedekind</a></div> <p>Hai valores que nun se pueden espresar como <a href="/wiki/N%C3%BAmberu_racional" title="Númberu racional">númberos racionales</a>, tal ye'l casu de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>. Sicasí ye claro que puede averase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> con númberos racionales tanto como se deseye. Podemos entós partir al conxuntu de los númberos racionales en dos subconxuntos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> de manera que nel conxuntu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> atópense tolos númberos racionales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6194bb47640c2314de8e34fafe66a3e2872cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.526ex; height:3.009ex;" alt="{\displaystyle x<{\sqrt {2}}}"></span> y en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> tolos númberos racionales tales que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759748cc47a3d9befa0bed12201ed82140e6acb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.526ex; height:3.009ex;" alt="{\displaystyle x>{\sqrt {2}}}"></span>. </p><p>Una <i>cortadura de dedekind</i> ye un <a href="/w/index.php?title=Par_orden%C3%A1u&action=edit&redlink=1" class="new" title="Par ordenáu (la páxina nun esiste)">par ordenáu</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce67314185650d6f0deba39db7dcec9378f4d4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}"></span> que fai precisamente esto. Conceptualmente, la cortadura ye'l "espaciu" qu'hai ente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. D'esta manera ye posible definir a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce67314185650d6f0deba39db7dcec9378f4d4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}"></span> tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{x\in \mathbb {Q} :x^{2}<2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{x\in \mathbb {Q} :x^{2}<2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575b21697a062e7f80d33b80a62128aa599d05b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.727ex; height:3.176ex;" alt="{\displaystyle A=\{x\in \mathbb {Q} :x^{2}<2\}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{x\in \mathbb {Q} :x^{2}>2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{x\in \mathbb {Q} :x^{2}>2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ba2f9a96b09e00c5eb31e50b2d37e10bac01ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.748ex; height:3.176ex;" alt="{\displaystyle B=\{x\in \mathbb {Q} :x^{2}>2\}}"></span>. </p><p>Ye posible demostrar que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> queda unívocamente definíu por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, d'esta manera la cortadura <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce67314185650d6f0deba39db7dcec9378f4d4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}"></span> amenórgase a cencielles a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p><p>Tamién ye demostrable que'l conxuntu de toles cortadures cumple colos axomes de los númberos reales, d'esta manera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> ye'l conxuntu de toles cortadures de Dedekind. Esta ye la primer construcción <a href="/w/index.php?title=Sistema_formal&action=edit&redlink=1" class="new" title="Sistema formal (la páxina nun esiste)">formal</a> de los númberos reales so la <a href="/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos">teoría de conxuntos</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Construcción_por_socesión_de_Cauchy"><span id="Construcci.C3.B3n_por_socesi.C3.B3n_de_Cauchy"></span>Construcción por socesión de Cauchy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=12" title="Editar seición: Construcción por socesión de Cauchy" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=12" title="Editar el código fuente de la sección: Construcción por socesión de Cauchy"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=Socesi%C3%B3n_de_Cauchy&action=edit&redlink=1" class="new" title="Socesión de Cauchy (la páxina nun esiste)">Socesión de Cauchy</a></div> <p>Les socesiones de Cauchy retomen la idea d'averar con númberos racionales un númberu real.<sup>[<i><a href="/wiki/Wikipedia:Verificabilid%C3%A1" title="Wikipedia:Verificabilidá">ensin referencies</a></i>]</sup> Tómese por casu, la igualdá. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\cfrac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\dots =\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\cfrac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\dots =\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e7a39f1235660fe67c13e940f9a1684da88bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.186ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\cfrac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\dots =\pi }"></span>.</dd></dl> <p>Ye claro qu'esta sumatoria opera namái colos númberos racionales de la forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {4(-1)^{n}}{2\,n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {4(-1)^{n}}{2\,n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9e29ec3a5f2f90b1ce1fc1bc2bc3df4e63c2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:7.997ex; height:7.176ex;" alt="{\displaystyle {\cfrac {4(-1)^{n}}{2\,n+1}}}"></span></dd></dl> <p>sicasí la resultancia final ye'l númberu irracional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64589f24cc8e145d021afa17d6564d55ea5a95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.719ex; height:1.676ex;" alt="{\displaystyle \pi \,}"></span>. Cada vez que s'añedir un términu, la espresión avérase más y más a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64589f24cc8e145d021afa17d6564d55ea5a95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.719ex; height:1.676ex;" alt="{\displaystyle \pi \,}"></span>. </p><p>Les socesiones de Cauchy xeneralicen esti conceutu pa definir a los númberos reales. Primero defínese qu'una <i>socesión de númberos racionales</i> ye una función se denota a cencielles por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span>. </p><p>Una <i>socesión de Cauchy</i> ye una socesión de númberos racionales onde los sos elementos cada vez son menos distintos. Más formalmente, defínese una <i>socesión de Cauchy</i> como una socesión de númberos racionales tales que pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon \in \mathbb {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \in \mathbb {Q} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3467694be00b116b5687866eb2c949fa5b375b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.104ex; height:2.843ex;" alt="{\displaystyle \epsilon \in \mathbb {Q} ^{+}}"></span> esiste un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1784c411bfa17be1e10d891101eb365daabda47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n_{0}\in \mathbb {N} }"></span> tal que pa tou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\geq {n_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n\geq {n_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c99236e18e5ea30c4c3e7bdac54b5e59a2b794b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.016ex; height:2.343ex;" alt="{\displaystyle m,n\geq {n_{0}}}"></span> cumplir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{m}-x_{n}|<\epsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{m}-x_{n}|<\epsilon \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19351db1af1d3fe7c3d8759a10650cab8c987f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.117ex; height:2.843ex;" alt="{\displaystyle |x_{m}-x_{n}|<\epsilon \,}"></span>. </p><p>D'esta manera ye posible definir al númberu real <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> como la socesión de númberos racionales: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\sum _{n=0}^{i}{\cfrac {4(-1)^{n}}{2n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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>i</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=\sum _{n=0}^{i}{\cfrac {4(-1)^{n}}{2n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a12ea375a7edce152b79733e2d458aae6d3b397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.967ex; height:7.176ex;" alt="{\displaystyle x_{i}=\sum _{n=0}^{i}{\cfrac {4(-1)^{n}}{2n+1}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Definición_de_los_númberos_reales"><span id="Definici.C3.B3n_de_los_n.C3.BAmberos_reales"></span>Definición de los númberos reales</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=13" title="Editar seición: Definición de los númberos reales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=13" title="Editar el código fuente de la sección: Definición de los númberos reales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sía Γ el conxuntu de les socesiones de Cauchy en Q. Sía la rellación siguiente, definida ente les socesiones de Cauchy de Q, (x<sub>n</sub>) y (y<sub>n</sub>): </p> <ul><li><center>(x<sub>n</sub>)ρ(y<sub>n</sub>) s. s.s. lim (x<sub>n</sub>-y<sub>n</sub>) = 0 cuando n → ∞ </center>.</li></ul> <ul><li>Esta rellación ρ ye una rellación d'equivalencia nel conxuntu de socesiones de Cauchy con elementos del conxuntu Q de los númberos racionales.</li></ul> <ul><li>Llamamos <b>conxuntu de los númberos reales</b> al conxuntu cociente <b> R = Γ/ρ</b>.</li> <li>Nel intre se define sobre R una llei de grupu aditivu, una rellación d'orde y una topoloxía. Demuéstrase que Q ( conxuntu de los racionales) ye isomorfu a una parte de R.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="Propiedá_Arquimediana_(Axoma_de_Arquímedes)"><span id="Propied.C3.A1_Arquimediana_.28Axoma_de_Arqu.C3.ADmedes.29"></span>Propiedá Arquimediana (Axoma de Arquímedes)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=14" title="Editar seición: Propiedá Arquimediana (Axoma de Arquímedes)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=14" title="Editar el código fuente de la sección: Propiedá Arquimediana (Axoma de Arquímedes)"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sían a > 0 y b númberos reales cualesquier, esiste un númberu natural n tal que na > b; esto espresa de la mesma que la socesión b/n tiende a cero.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Operaciones_con_númberos_reales"><span id="Operaciones_con_n.C3.BAmberos_reales"></span>Operaciones con númberos reales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=15" title="Editar seición: Operaciones con númberos reales" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=15" title="Editar el código fuente de la sección: Operaciones con númberos reales"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Con númberos reales pueden realizase tou tipu d'operaciones básiques con diverses esceiciones importantes: </p> <ol><li>Nun esisten <a href="/w/index.php?title=Radicaci%C3%B3n&action=edit&redlink=1" class="new" title="Radicación (la páxina nun esiste)">raigaños</a> d'orde par (cuadraes, cuartes, sestes, etc.) de númberos negativos en númberos reales, (anque sí esisten nel conxuntu de los <a href="/wiki/N%C3%BAmberu_complexu" title="Númberu complexu">númberos complexos</a> onde diches operaciones sí tán definíes).</li> <li>La <a href="/w/index.php?title=Divisi%C3%B3n_ente_cero&action=edit&redlink=1" class="new" title="División ente cero (la páxina nun esiste)">división ente cero</a> nun ta definida (pos cero nun tener inversu multiplicativu, esto ye, nun esiste númberu <i>x</i> tal que 0·<i>x</i>=1).</li> <li>Nun puede topase el llogaritmu d'un númberu real negativu, cualesquier sía la base de llogaritmos, un númberu positivu distintu de 1.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ol> <p>Estes restricciones tienen repercusiones n'otres árees de les matemátiques como'l cálculu: esisten <a href="/w/index.php?title=As%C3%ADntotas&action=edit&redlink=1" class="new" title="Asíntotas (la páxina nun esiste)">asíntotas</a> verticales nos llugares onde'l denominador d'una <a href="/wiki/Funci%C3%B3n_(matem%C3%A1tica)" class="mw-redirect" title="Función (matemática)">función</a> racional tiende a cero, esto ye, naquellos valores de la variable nos que se presentaría una <a href="/w/index.php?title=Divisi%C3%B3n_ente_cero&action=edit&redlink=1" class="new" title="División ente cero (la páxina nun esiste)">división ente cero</a>, o nun esiste gráfica real naquellos valores de la variable en que resulten númberos negativos pa raigaños d'orde par, por mentar un exemplu de construcción de gráfiques en xeometría analítica. </p> <div class="mw-heading mw-heading2"><h2 id="Dos_particiones">Dos particiones</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=16" title="Editar seición: Dos particiones" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=16" title="Editar el código fuente de la sección: Dos particiones"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>El conxuntu de los reales ye la unión dixunta de los racionales y de los irracionales</li> <li>El conxuntu R ye la unión d'A y T, Al conxuntu de los reales alxebraicos y T el conxuntu de los trascendentes<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=17" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=17" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Plant%C3%ADa:Clasificaci%C3%B3n_n%C3%BAmberu&action=edit&redlink=1" class="new" title="Plantía:Clasificación númberu (la páxina nun esiste)">Plantía:Clasificación númberu</a> </p> <div class="mw-heading mw-heading3"><h3 id="Dos_clasificaciones">Dos clasificaciones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=18" title="Editar seición: Dos clasificaciones" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=18" title="Editar el código fuente de la sección: Dos clasificaciones"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Hai una partición del conxuntu de los reales en dos subconxuntos: racionales ya irracionales. Tolos racionales son alxebraicos y los irracionales pueden ser alxebraicos y trascendentes.</li> <li>Hai otra partición del conxuntu de los reales n'otros dos subconxuntos: alxebraicos y trascendentes. Los primeres son racionales ya irracionales. Tolos trascendentes son irracionales<sup id="cite_ref-Tsipkin_2-2" class="reference"><a href="#cite_note-Tsipkin-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="Notes_y_referencies">Notes y referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=19" title="Editar seición: Notes y referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=19" title="Editar el código fuente de la sección: Notes y referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-in-1"><span class="mw-cite-backlink"><a href="#cite_ref-in_1-0">↑</a></span> <span class="reference-text"><cite style="font-style:normal">Arias Cabezas, José María;  Maza Sáez, Ildefonso (2008). «Aritmética y Álxebra», <i>Matemátiques 1</i>. Madrid: Grupu Editorial Bruño, Sociedá Llindada, páx. 13. <a href="/wiki/Especial:FuentesDeLibros/9788421659854" title="Especial:FuentesDeLibros/9788421659854">ISBN 9788421659854</a>.</cite></span> </li> <li id="cite_note-Tsipkin-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Tsipkin_2-0">2,0</a></sup> <sup><a href="#cite_ref-Tsipkin_2-1">2,1</a></sup> <sup><a href="#cite_ref-Tsipkin_2-2">2,2</a></sup></span> <span class="reference-text"><i>Manual de matemátiques</i> (1985) Tsipkin, Editorial Mir, Moscú, traducción de Shapovalova; pg. 86</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite style="font-style:normal" id="Referencia-Anglin-1991">Anglin, W. S. (1991). <i>Mathematics: A concise history and philosophy</i>. Springer. <a href="/wiki/Especial:FuentesDeLibros/3-540-94280-7" title="Especial:FuentesDeLibros/3-540-94280-7">ISBN 3-540-94280-7</a>.</cite></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite style="font-style:normal" id="Referencia-Dantzig-1955">Dantzig, Tobias (1955). <i>The Bequest of the Greeks</i>. London: Unwin Brothers LTD. 3982581.</cite></span> </li> <li id="cite_note-Stillwell-5"><span class="mw-cite-backlink"><a href="#cite_ref-Stillwell_5-0">↑</a></span> <span class="reference-text"><cite style="font-style:normal" id="Referencia-Stillwell-1989">Stillwell, John (1989). <i>Mathematics and its History</i>. Springer-Verlag. 19269766. <a href="/wiki/Especial:FuentesDeLibros/3-540-96981-0" title="Especial:FuentesDeLibros/3-540-96981-0">ISBN 3-540-96981-0</a>.</cite></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Haaser y otros, Kudiatsev; Bartle y otru, siguen</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">"El conceutu de númberu de Númberu" (1973) César Trejo. La propuesta ye de D. Hilbert qu'apaeció nel so célebre artículu en 1900: <i>Über die Zahlbegriff</i> páxs. 82 y 83</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Zamansky. <i>Introducción a la álxebra y analís modernu</i>. Montaner y Simon, Barcelona</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Haaser y otros: Análisi matemáticu I</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Aplíquese la definición de llogaritmu</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Courant: ¿Qué ye la matemática?</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Enllaces_esternos">Enllaces esternos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmberu_real&veaction=edit&section=20" title="Editar seición: Enllaces esternos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=N%C3%BAmberu_real&action=edit&section=20" title="Editar el código fuente de la sección: Enllaces esternos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="plainlinks commons navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=ast">Wikimedia Commons</a> acueye conteníu multimedia sobre <i><b><a href="https://commons.wikimedia.org/wiki/Category:Real_numbers" class="extiw" title="commons:Category:Real numbers">Númberu real</a></b></i>.</span></li></ul> <ul><li>Weisstein, Eric W. «<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RealNumber.html">Númberu real</a>» <span style="color:var(--color-subtle, #54595d);"><span style="color:var(--color-subtle, #54595d)">(inglés)</span></span>. <i><a href="/w/index.php?title=MathWorld&action=edit&redlink=1" class="new" title="MathWorld (la páxina nun esiste)">MathWorld</a></i>.  <a href="/w/index.php?title=Wolfram_Research&action=edit&redlink=1" class="new" title="Wolfram Research (la páxina nun esiste)">Wolfram Research</a>.</li></ul> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r2260362">.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid #c8ccd1;background-color:#eaecf0;color:#222222}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control navigation-not-searchable"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r4075543">.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:r4182964">.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}</style></div><div role="navigation" class="navbox" aria-labelledby="Control_d&#039;autoridaes" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_d&#039;autoridaes" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Ayuda:Control_d%27autoridaes" title="Ayuda:Control d'autoridaes">Control d'autoridaes</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><b>Proyeutos Wikimedia</b></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q12916" class="extiw" title="wikidata:Q12916">Q12916</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Real_numbers">Real numbers</a></span></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioth%C3%A8que_nationale_de_France" class="mw-redirect" title="Bibliothèque nationale de France">BNF</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11977586x">11977586x</a> <a rel="nofollow" class="external text" href="http://data.bnf.fr/ark:/12148/cb11977586x">(data)</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Integrated_Authority_File" class="mw-redirect" title="Integrated Authority File">GND</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4202628-3">4202628-3</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Library_of_Congress_Control_Number" class="mw-redirect" title="Library of Congress Control Number">LCCN</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85093221">sh85093221</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00574870">00574870</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Library_of_the_Czech_Republic" class="mw-redirect" title="National Library of the Czech Republic">NKC</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph125164">ph125164</a></span></li> <li><b>Diccionarios y enciclopedies</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclopedia_Brit%C3%A1nica" class="mw-redirect" title="Enciclopedia Británica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/real-number">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q12916" class="extiw" title="wikidata:Q12916">Q12916</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Real_numbers">Real numbers</a></span></span></li></ul> </div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Sacáu de «<a dir="ltr" href="https://ast.wikipedia.org/w/index.php?title=Númberu_real&oldid=4178560">https://ast.wikipedia.org/w/index.php?title=Númberu_real&oldid=4178560</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categor%C3%ADas" title="Especial:Categorías">Categoríes</a>: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_plant%C3%ADes_de_notes_d%27encabezamientu_enllaciando_a_p%C3%A1xines_que_nun_esisten" title="Categoría:Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten">Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Revisar_traducci%C3%B3n" title="Categoría:Wikipedia:Revisar traducción">Wikipedia:Revisar traducción</a></li><li><a href="/wiki/Categor%C3%ADa:N%C3%BAmberos_reales" title="Categoría:Númberos reales">Númberos reales</a></li><li><a href="/wiki/Categor%C3%ADa:Matem%C3%A1tica_elemental" title="Categoría:Matemática elemental">Matemática elemental</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoríes anubríes: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_qu%27usen_ficha_ensin_datos_en_fileres" title="Categoría:Wikipedia:Artículos qu'usen ficha ensin datos en fileres">Wikipedia:Artículos qu'usen ficha ensin datos en fileres</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_datos_ensin_referencies" title="Categoría:Wikipedia:Artículos con datos ensin referencies">Wikipedia:Artículos con datos ensin referencies</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:P%C3%A1xines_con_enllaz_commonscat_igual_que_Wikidata" title="Categoría:Wikipedia:Páxines con enllaz commonscat igual que Wikidata">Wikipedia:Páxines con enllaz commonscat igual que Wikidata</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_identificadores_BNF" title="Categoría:Wikipedia:Artículos con identificadores BNF">Wikipedia:Artículos con identificadores BNF</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_identificadores_GND" title="Categoría:Wikipedia:Artículos con identificadores GND">Wikipedia:Artículos con identificadores GND</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_identificadores_LCCN" title="Categoría:Wikipedia:Artículos con identificadores LCCN">Wikipedia:Artículos con identificadores LCCN</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Tradubot" title="Categoría:Wikipedia:Tradubot">Wikipedia:Tradubot</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"> La última edición d'esta páxina foi'l 8 abr 2024 a les 16:26.</li> <li id="footer-info-copyright">El testu ta disponible baxo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.es">Llicencia Creative Commons Reconocimientu/CompartirIgual 4.0</a>; puen aplicase términos adicionales. Llei <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">les condiciones d'usu</a> pa más detalles.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/es">Política d'intimidá</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Tocante_a">Tocante a Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Avisu_xeneral">Avisu llegal</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Códigu de conducta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desendolcadores</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/ast.wikipedia.org">Estadístiques</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement/es">Declaración de cookies</a></li> <li id="footer-places-mobileview"><a href="//ast.m.wikipedia.org/w/index.php?title=N%C3%BAmberu_real&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Vista pa móvil</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-6zb5l","wgBackendResponseTime":142,"wgPageParseReport":{"limitreport":{"cputime":"0.440","walltime":"0.983","ppvisitednodes":{"value":2942,"limit":1000000},"postexpandincludesize":{"value":19328,"limit":2097152},"templateargumentsize":{"value":3806,"limit":2097152},"expansiondepth":{"value":18,"limit":100},"expensivefunctioncount":{"value":10,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":19595,"limit":5000000},"entityaccesscount":{"value":8,"limit":400},"timingprofile":["100.00% 453.041 1 -total"," 36.89% 167.113 1 Plantía:Control_d'autoridaes"," 27.07% 122.652 1 Plantía:Ficha_xenérica"," 26.35% 119.356 1 Plantía:Infobox"," 6.23% 28.208 1 Plantía:Commonscat"," 6.00% 27.199 3 Plantía:AP"," 5.33% 24.145 1 Plantía:MathWorld"," 4.67% 21.150 1 Plantía:Cita_web"," 4.30% 19.472 1 Plantía:Llistaref"," 4.15% 18.782 1 Plantía:Commonscat/categoríes"]},"scribunto":{"limitreport-timeusage":{"value":"0.184","limit":"10.000"},"limitreport-memusage":{"value":5331145,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-5dc468848-ft7q7","timestamp":"20241122123008","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"N\u00famberu real","url":"https:\/\/ast.wikipedia.org\/wiki\/N%C3%BAmberu_real","sameAs":"http:\/\/www.wikidata.org\/entity\/Q12916","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q12916","author":{"@type":"Organization","name":"Collaboradores de los proyeutos de Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2018-06-17T17:13:39Z","dateModified":"2024-04-08T16:26:58Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/4\/47\/N%C3%BAmeros_reales.svg","headline":"n\u00famberu que puede representase por una parte entera y una llista finita o infinita de decimales"}</script> </body> </html>