Pi — Wikipèdia

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tavèrna</span></a></li><li id="n-thema" class="mw-list-item"><a href="/wiki/Portal:Acu%C3%A8lh"><span>Portals tematics</span></a></li><li id="n-alphindex" class="mw-list-item"><a href="/wiki/Especial:Totas_las_paginas"><span>Indèx alfabetic</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Pagina_a_l%27azard" title="Afichar una pagina a l'azard [x]" accesskey="x"><span>Una pagina a l'azard</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Contact-url"><span>contact</span></a></li> </ul> </div> </div> <div id="p-Contribuir" class="vector-menu mw-portlet mw-portlet-Contribuir" > <div class="vector-menu-heading"> Contribuir </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Ajuda:Somari" title="L'endrech per s'assabentar."><span>Ajuda</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikip%C3%A8dia:Acu%C3%A8lh" title="A prepaus del projècte, çò que podètz far, ont trobar d'informacions"><span>Comunautat</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Darri%C3%A8rs_cambiaments" title="Lista dels darrièrs cambiaments sul wiki [r]" accesskey="r"><span>Darrièrs cambiaments</span></a></li><li id="n-aboutwp" class="mw-list-item"><a href="/wiki/Wikip%C3%A8dia:Acu%C3%A8lh_dels_nov%C3%A8ls_venguts"><span>Acuèlh dels novèls venguts</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Acu%C3%A8lh" 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="Wikipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-oc.svg" style="width: 7.5em; height: 1.1875em;"> <img class="mw-logo-tagline" alt="l'enciclopèdia liura" src="/static/images/mobile/copyright/wikipedia-tagline-oc.svg" width="120" 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class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Modifier l'apparence de la taille, de la largeur et de la couleur de la police de la page" > <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="Aparença" > <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">Aparença</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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_oc.wikipedia.org&uselang=oc" class=""><span>Far un don</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_un_compte&returnto=Pi" title="Vos es conselhat de crear un compte e de vos connectar ; pasmens, es pas obligatòri" class=""><span>Crear un compte</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:Nom_d%27utilizaire&returnto=Pi" title="Sètz convidat(ada) a vos identificar, mas es pas obligatòri. [o]" accesskey="o" class=""><span>Se connectar</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="Més opcions" > <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="Aisinas personalas" > <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">Aisinas personalas</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 d'usuari" > <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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_oc.wikipedia.org&uselang=oc"><span>Far un don</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_un_compte&returnto=Pi" title="Vos es conselhat de crear un compte e de vos connectar ; pasmens, es pas obligatòri"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Nom_d%27utilizaire&returnto=Pi" title="Sètz convidat(ada) a vos identificar, mas es pas obligatòri. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Se connectar</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"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</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:Mas_contribucions" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Mas_discussions" title="La pagina de discussion per aquesta adreça IP [n]" accesskey="n"><span>Discussion</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="Lloc"> <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="Somari" 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">Somari</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">Escondre</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">Començament</div> </a> </li> <li id="toc-Definicions_e_premierei_proprietats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicions_e_premierei_proprietats"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicions e premierei proprietats</span> </div> </a> <button aria-controls="toc-Definicions_e_premierei_proprietats-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>Commuta la subsecció Definicions e premierei proprietats</span> </button> <ul id="toc-Definicions_e_premierei_proprietats-sublist" class="vector-toc-list"> <li id="toc-Definicions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definicions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definicions</span> </div> </a> <ul id="toc-Definicions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irracionalitat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irracionalitat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Irracionalitat</span> </div> </a> <ul id="toc-Irracionalitat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transcendéncia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transcendéncia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Transcendéncia</span> </div> </a> <ul id="toc-Transcendéncia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representacion_decimala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representacion_decimala"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Representacion decimala</span> </div> </a> <ul id="toc-Representacion_decimala-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aproximacion_de_π" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aproximacion_de_π"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Aproximacion de π</span> </div> </a> <ul id="toc-Aproximacion_de_π-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Utilizacion_en_matematicas_e_en_sciéncias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Utilizacion_en_matematicas_e_en_sciéncias"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Utilizacion en matematicas e en sciéncias</span> </div> </a> <button aria-controls="toc-Utilizacion_en_matematicas_e_en_sciéncias-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>Commuta la subsecció Utilizacion en matematicas e en sciéncias</span> </button> <ul id="toc-Utilizacion_en_matematicas_e_en_sciéncias-sublist" class="vector-toc-list"> <li id="toc-Geometria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometria"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Geometria</span> </div> </a> <ul id="toc-Geometria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autrei_definicions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Autrei_definicions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Autrei definicions</span> </div> </a> <ul id="toc-Autrei_definicions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seguidas_e_serias" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seguidas_e_serias"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Seguidas e serias</span> </div> </a> <ul id="toc-Seguidas_e_serias-sublist" class="vector-toc-list"> <li id="toc-Metòde_d’Arquimèdes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Metòde_d’Arquimèdes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Metòde d’Arquimèdes</span> </div> </a> <ul id="toc-Metòde_d’Arquimèdes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Somas_e_produchs_infinits" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Somas_e_produchs_infinits"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Somas e produchs infinits</span> </div> </a> <ul id="toc-Somas_e_produchs_infinits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seguidas_recursivas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Seguidas_recursivas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Seguidas recursivas</span> </div> </a> <ul id="toc-Seguidas_recursivas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Foncion_zêta_de_Riemann" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Foncion_zêta_de_Riemann"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.4</span> <span>Foncion zêta de Riemann</span> </div> </a> <ul id="toc-Foncion_zêta_de_Riemann-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Foncion_Gamma_d’Euler" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Foncion_Gamma_d’Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.5</span> <span>Foncion Gamma d’Euler</span> </div> </a> <ul id="toc-Foncion_Gamma_d’Euler-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Probabilitats_e_estatisticas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probabilitats_e_estatisticas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Probabilitats e estatisticas</span> </div> </a> <ul id="toc-Probabilitats_e_estatisticas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seguida_logistica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seguida_logistica"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Seguida logistica</span> </div> </a> <ul id="toc-Seguida_logistica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proprietats_avançadas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietats_avançadas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Proprietats avançadas</span> </div> </a> <button aria-controls="toc-Proprietats_avançadas-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>Commuta la subsecció Proprietats avançadas</span> </button> <ul id="toc-Proprietats_avançadas-sublist" class="vector-toc-list"> <li id="toc-Aproximacions_numericas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aproximacions_numericas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Aproximacions numericas</span> </div> </a> <ul id="toc-Aproximacions_numericas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fraccions_continuas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fraccions_continuas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Fraccions continuas</span> </div> </a> <ul id="toc-Fraccions_continuas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Questions_dubèrtas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Questions_dubèrtas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Questions dubèrtas</span> </div> </a> <ul id="toc-Questions_dubèrtas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lo_nombre_π_dins_l'art" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lo_nombre_π_dins_l'art"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Lo nombre π dins l'art</span> </div> </a> <ul id="toc-Lo_nombre_π_dins_l'art-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nòtas_e_referéncias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nòtas_e_referéncias"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Nòtas e referéncias</span> </div> </a> <button aria-controls="toc-Nòtas_e_referéncias-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>Commuta la subsecció Nòtas e referéncias</span> </button> <ul id="toc-Nòtas_e_referéncias-sublist" class="vector-toc-list"> <li id="toc-Liames_extèrnes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liames_extèrnes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Liames extèrnes</span> </div> </a> <ul id="toc-Liames_extèrnes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Somari" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Pi</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="Vés a un article en una altra llengua. Disponible en 162 llengües" > <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-162" 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">162 lengas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://af.wikipedia.org/wiki/Pi" title="Pi - afrikaans" lang="af" hreflang="af" data-title="Pi" 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/Pi_(Mathematik)" title="Pi (Mathematik) - aleman de Soïssa" lang="gsw" hreflang="gsw" data-title="Pi (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="aleman de Soïssa" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%93%E1%8B%AD" title="ፓይ - amaric" lang="am" hreflang="am" data-title="ፓይ" data-language-autonym="አማርኛ" data-language-local-name="amaric" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_%CF%80" title="Numero π - aragonés" lang="an" hreflang="an" data-title="Numero π" data-language-autonym="Aragonés" data-language-local-name="aragonés" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="ط (رياضيات) - arabi" lang="ar" hreflang="ar" data-title="ط (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabi" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A8%D8%A7%D9%89_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="باى (رياضيات) - àrab egipci" lang="arz" hreflang="arz" data-title="باى (رياضيات)" data-language-autonym="مصرى" data-language-local-name="àrab egipci" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%87" title="পাই - assamès" lang="as" hreflang="as" data-title="পাই" data-language-autonym="অসমীয়া" data-language-local-name="assamès" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_%CF%80" title="Númberu π - asturian" lang="ast" hreflang="ast" data-title="Númberu π" data-language-autonym="Asturianu" data-language-local-name="asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Pi" title="Pi - azerbaijani" lang="az" hreflang="az" data-title="Pi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%BE%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%E2%80%8C%D8%B3%DB%8C" title="پی سایی‌سی - South Azerbaijani" lang="azb" hreflang="azb" data-title="پی سایی‌سی" 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%9F%D0%B8_(%D2%BB%D0%B0%D0%BD)" title="Пи (һан) - baixkir" lang="ba" hreflang="ba" data-title="Пи (һан)" data-language-autonym="Башҡортса" data-language-local-name="baixkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Pi" title="Pi - samogitien" lang="sgs" hreflang="sgs" data-title="Pi" data-language-autonym="Žemaitėška" data-language-local-name="samogitien" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Pi" title="Pi - Central Bikol" lang="bcl" hreflang="bcl" data-title="Pi" 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%9F%D1%96" title="Пі - belarús" lang="be" hreflang="be" data-title="Пі" data-language-autonym="Беларуская" data-language-local-name="belarús" 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%9F%D1%96" title="Пі - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Пі" 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%9F%D0%B8" title="Пи - bulgar" lang="bg" hreflang="bg" data-title="Пи" data-language-autonym="Български" data-language-local-name="bulgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Pi" title="Pi - banjar" lang="bjn" hreflang="bjn" data-title="Pi" data-language-autonym="Banjar" data-language-local-name="banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%87" title="পাই - bengalin" lang="bn" hreflang="bn" data-title="পাই" data-language-autonym="বাংলা" data-language-local-name="bengalin" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Pi_(niver)" title="Pi (niver) - breton" lang="br" hreflang="br" data-title="Pi (niver)" data-language-autonym="Brezhoneg" data-language-local-name="breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Pi" title="Pi - bosniac" lang="bs" hreflang="bs" data-title="Pi" data-language-autonym="Bosanski" data-language-local-name="bosniac" 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%9F%D0%B8_(%D1%82%D0%BE%D0%BE)" title="Пи (тоо) - Russia Buriat" lang="bxr" hreflang="bxr" data-title="Пи (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-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://ca.wikipedia.org/wiki/Nombre_%CF%80" title="Nombre π - catalan" lang="ca" hreflang="ca" data-title="Nombre π" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/I%C3%A8ng-ci%C5%AD-l%E1%B9%B3%CC%86k" title="Ièng-ciŭ-lṳ̆k - Mindong" lang="cdo" hreflang="cdo" data-title="Ièng-ciŭ-lṳ̆k" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Pi" title="Pi - cebuan" lang="ceb" hreflang="ceb" data-title="Pi" data-language-autonym="Cebuano" data-language-local-name="cebuan" class="interlanguage-link-target"><span>Cebuano</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%D8%A7%DB%8C" title="پای - kurd central" lang="ckb" hreflang="ckb" data-title="پای" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="bons articles"><a href="https://cs.wikipedia.org/wiki/P%C3%AD_(%C4%8D%C3%ADslo)" title="Pí (číslo) - chèc" lang="cs" hreflang="cs" data-title="Pí (číslo)" data-language-autonym="Čeština" data-language-local-name="chèc" 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%9F%D0%B8_(%D1%85%D0%B8%D1%81%D0%B5%D0%BF)" title="Пи (хисеп) - chovash" lang="cv" hreflang="cv" data-title="Пи (хисеп)" data-language-autonym="Чӑвашла" data-language-local-name="chovash" 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/Pi_(mathemateg)" title="Pi (mathemateg) - gal·lès" lang="cy" hreflang="cy" data-title="Pi (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" 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/Pi_(tal)" title="Pi (tal) - danés" lang="da" hreflang="da" data-title="Pi (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 badge-Q17437798 badge-goodarticle mw-list-item" title="bons articles"><a href="https://de.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl - alemand" lang="de" hreflang="de" data-title="Kreiszahl" data-language-autonym="Deutsch" data-language-local-name="alemand" 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/Amar_Pi" title="Amar Pi - Zazaki" lang="diq" hreflang="diq" data-title="Amar Pi" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Konstanta_%CF%80" title="Konstanta π - baix sòrab" lang="dsb" hreflang="dsb" data-title="Konstanta π" data-language-autonym="Dolnoserbski" data-language-local-name="baix sòrab" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%84%CE%B1%CE%B8%CE%B5%CF%81%CE%AC)" title="Π (μαθηματική σταθερά) - grèc" lang="el" hreflang="el" data-title="Π (μαθηματική σταθερά)" data-language-autonym="Ελληνικά" data-language-local-name="grèc" 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/Pi_gr%C4%93c" title="Pi grēc - Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Pi grēc" 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 badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://en.wikipedia.org/wiki/Pi" title="Pi - anglés" lang="en" hreflang="en" data-title="Pi" data-language-autonym="English" data-language-local-name="anglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://eo.wikipedia.org/wiki/Pi_(nombro)" title="Pi (nombro) - esperanto" lang="eo" hreflang="eo" data-title="Pi (nombro)" 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_%CF%80" title="Número π - espanhòl" lang="es" hreflang="es" data-title="Número π" data-language-autonym="Español" data-language-local-name="espanhòl" 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/Pii" title="Pii - estonian" lang="et" hreflang="et" data-title="Pii" data-language-autonym="Eesti" data-language-local-name="estonian" 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/Pi_(zenbakia)" title="Pi (zenbakia) - basc" lang="eu" hreflang="eu" data-title="Pi (zenbakia)" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/N%C3%BAmiru_%CF%80" title="Númiru π - extremeny" lang="ext" hreflang="ext" data-title="Númiru π" data-language-autonym="Estremeñu" data-language-local-name="extremeny" class="interlanguage-link-target"><span>Estremeñu</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_%D9%BE%DB%8C" title="عدد پی - perse" lang="fa" hreflang="fa" data-title="عدد پی" data-language-autonym="فارسی" data-language-local-name="perse" 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/Pii_(vakio)" title="Pii (vakio) - finlandés" lang="fi" hreflang="fi" data-title="Pii (vakio)" 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/Pii" title="Pii - võro" lang="vro" hreflang="vro" data-title="Pii" data-language-autonym="Võro" data-language-local-name="võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Pi" title="Pi - fijian" lang="fj" hreflang="fj" data-title="Pi" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Pi" title="Pi - feroés" lang="fo" hreflang="fo" data-title="Pi" 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/Pi" title="Pi - francés" lang="fr" hreflang="fr" data-title="Pi" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr badge-Q70894304 mw-list-item" title=""><a href="https://frr.wikipedia.org/wiki/Pi" title="Pi - frisó septentrional" lang="frr" hreflang="frr" data-title="Pi" data-language-autonym="Nordfriisk" data-language-local-name="frisó septentrional" 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/Pi_gr%C3%AAc" title="Pi grêc - friolan" lang="fur" hreflang="fur" data-title="Pi grêc" data-language-autonym="Furlan" data-language-local-name="friolan" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Py_(wiskunde)" title="Py (wiskunde) - frisó occidental" lang="fy" hreflang="fy" data-title="Py (wiskunde)" data-language-autonym="Frysk" data-language-local-name="frisó occidental" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/P%C3%AD_(uimhir)" title="Pí (uimhir) - irlandés" lang="ga" hreflang="ga" data-title="Pí (uimhir)" 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%9C%93%E5%91%A8%E7%8E%87" title="圓周率 - xinès gan" lang="gan" hreflang="gan" data-title="圓周率" data-language-autonym="贛語" data-language-local-name="xinès 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/Pi" title="Pi - créole guyanais" lang="gcr" hreflang="gcr" data-title="Pi" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="créole guyanais" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Pi_(%C3%A0ireamh)" title="Pi (àireamh) - gaelic escossés" lang="gd" hreflang="gd" data-title="Pi (àireamh)" data-language-autonym="Gàidhlig" data-language-local-name="gaelic escossés" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_pi" title="Número pi - galician" lang="gl" hreflang="gl" data-title="Número pi" data-language-autonym="Galego" data-language-local-name="galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Pi" title="Pi - guaraní" lang="gn" hreflang="gn" data-title="Pi" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraní" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AA%BE%E0%AA%87" title="પાઇ - gujarati" lang="gu" hreflang="gu" data-title="પાઇ" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Pi" title="Pi - haussa" lang="ha" hreflang="ha" data-title="Pi" data-language-autonym="Hausa" data-language-local-name="haussa" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Pai_(makemakika)" title="Pai (makemakika) - hawaià" lang="haw" hreflang="haw" data-title="Pai (makemakika)" data-language-autonym="Hawaiʻi" data-language-local-name="hawaià" class="interlanguage-link-target"><span>Hawaiʻi</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%90%D7%99" title="פאי - ebrèu" lang="he" hreflang="he" data-title="פאי" data-language-autonym="עברית" data-language-local-name="ebrèu" 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%AA%E0%A4%BE%E0%A4%88" title="पाई - Indi" lang="hi" hreflang="hi" data-title="पाई" data-language-autonym="हिन्दी" data-language-local-name="Indi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Pi" title="Pi - hindi de Fiji" lang="hif" hreflang="hif" data-title="Pi" data-language-autonym="Fiji Hindi" data-language-local-name="hindi de Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pi_(broj)" title="Pi (broj) - croat" lang="hr" hreflang="hr" data-title="Pi (broj)" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Konstanta_%CF%80" title="Konstanta π - naut sorab" lang="hsb" hreflang="hsb" data-title="Konstanta π" data-language-autonym="Hornjoserbsce" data-language-local-name="naut sorab" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Pi_(matematik)" title="Pi (matematik) - crioll d’Haití" lang="ht" hreflang="ht" data-title="Pi (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="crioll d’Haití" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pi_(sz%C3%A1m)" title="Pi (szám) - ongrés" lang="hu" hreflang="hu" data-title="Pi (szám)" data-language-autonym="Magyar" data-language-local-name="ongrés" 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/%D5%8A%D5%AB_%D5%A9%D5%AB%D5%BE" title="Պի թիվ - armèni" lang="hy" hreflang="hy" data-title="Պի թիվ" data-language-autonym="Հայերեն" data-language-local-name="armèni" 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/Pi" title="Pi - interlingua" lang="ia" hreflang="ia" data-title="Pi" 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/Pi" title="Pi - iban" lang="iba" hreflang="iba" data-title="Pi" 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/Pi" title="Pi - indonesian" lang="id" hreflang="id" data-title="Pi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Pi" title="Pi - ido" lang="io" hreflang="io" data-title="Pi" 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/P%C3%AD" title="Pí - islandés" lang="is" hreflang="is" data-title="Pí" 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/Pi_greco" title="Pi greco - italian" lang="it" hreflang="it" data-title="Pi greco" data-language-autonym="Italiano" data-language-local-name="italian" 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%86%86%E5%91%A8%E7%8E%87" title="円周率 - japonés" lang="ja" hreflang="ja" data-title="円周率" data-language-autonym="日本語" data-language-local-name="japonés" 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/Pi" title="Pi - crioll anglès de Jamaica" lang="jam" hreflang="jam" data-title="Pi" data-language-autonym="Patois" data-language-local-name="crioll anglès de Jamaica" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pi" title="Pi - javanés" lang="jv" hreflang="jv" data-title="Pi" data-language-autonym="Jawa" data-language-local-name="javanés" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%98_(%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98)" title="პი (რიცხვი) - georgian" lang="ka" hreflang="ka" data-title="პი (რიცხვი)" data-language-autonym="ქართული" data-language-local-name="georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Pi" title="Pi - cabil" lang="kab" hreflang="kab" data-title="Pi" data-language-autonym="Taqbaylit" data-language-local-name="cabil" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%81%D0%B0%D0%BD)" title="Пи (сан) - cazac" lang="kk" hreflang="kk" data-title="Пи (сан)" data-language-autonym="Қазақша" data-language-local-name="cazac" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%88" title="ಪೈ - kannada" lang="kn" hreflang="kn" data-title="ಪೈ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko badge-Q17437798 badge-goodarticle mw-list-item" title="bons articles"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율 - corean" lang="ko" hreflang="ko" data-title="원주율" data-language-autonym="한국어" data-language-local-name="corean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ksh mw-list-item"><a href="https://ksh.wikipedia.org/wiki/Pi_(Kr%C3%A4j%C3%9Fzal)" title="Pi (Kräjßzal) - kölsch" lang="ksh" hreflang="ksh" data-title="Pi (Kräjßzal)" data-language-autonym="Ripoarisch" data-language-local-name="kölsch" class="interlanguage-link-target"><span>Ripoarisch</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Pi" title="Pi - curd" lang="ku" hreflang="ku" data-title="Pi" data-language-autonym="Kurdî" data-language-local-name="curd" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Pi" title="Pi - cornic" lang="kw" hreflang="kw" data-title="Pi" data-language-autonym="Kernowek" data-language-local-name="cornic" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи - kirguís" lang="ky" hreflang="ky" data-title="Пи" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://la.wikipedia.org/wiki/Numerus_pi" title="Numerus pi - latin" lang="la" hreflang="la" data-title="Numerus pi" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Pi_(Zuel)" title="Pi (Zuel) - luxemborgés" lang="lb" hreflang="lb" data-title="Pi (Zuel)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemborgés" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="Пи (число) - lesguià" lang="lez" hreflang="lez" data-title="Пи (число)" data-language-autonym="Лезги" data-language-local-name="lesguià" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Pi_(wisk%C3%B3nde)" title="Pi (wiskónde) - limburguès" lang="li" hreflang="li" data-title="Pi (wiskónde)" data-language-autonym="Limburgs" data-language-local-name="limburguès" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Pi_gregh" title="Pi gregh - llombard" lang="lmo" hreflang="lmo" data-title="Pi gregh" data-language-autonym="Lombard" data-language-local-name="llombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pi" title="Pi - lituan" lang="lt" hreflang="lt" data-title="Pi" data-language-autonym="Lietuvių" data-language-local-name="lituan" 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/P%C4%AB" title="Pī - leton" lang="lv" hreflang="lv" data-title="Pī" data-language-autonym="Latviešu" data-language-local-name="leton" 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/Pi" title="Pi - malgash" lang="mg" hreflang="mg" data-title="Pi" data-language-autonym="Malagasy" data-language-local-name="malgash" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Pi" title="Pi - minangkabau" lang="min" hreflang="min" data-title="Pi" data-language-autonym="Minangkabau" data-language-local-name="minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи - macedonian" lang="mk" hreflang="mk" data-title="Пи" data-language-autonym="Македонски" data-language-local-name="macedonian" 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%AA%E0%B5%88_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="പൈ (ഗണിതം) - malaiàlam" lang="ml" hreflang="ml" data-title="പൈ (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи - mongòl" lang="mn" hreflang="mn" data-title="Пи" data-language-autonym="Монгол" data-language-local-name="mongòl" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%AF_(%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%B0%E0%A4%BE%E0%A4%82%E0%A4%95)" title="पाय (स्थिरांक) - marathi" lang="mr" hreflang="mr" data-title="पाय (स्थिरांक)" 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/Pi" title="Pi - malai" lang="ms" hreflang="ms" data-title="Pi" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" 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%95%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA_(%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC)" title="ပိုင် (သင်္ချာ) - birman" lang="my" hreflang="my" data-title="ပိုင် (သင်္ချာ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krinktall" title="Krinktall - baix alemany" lang="nds" hreflang="nds" data-title="Krinktall" data-language-autonym="Plattdüütsch" data-language-local-name="baix alemany" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%88" title="पाई - nepalés" lang="ne" hreflang="ne" data-title="पाई" data-language-autonym="नेपाली" data-language-local-name="nepalés" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%87" title="पाइ - newari" lang="new" hreflang="new" data-title="पाइ" data-language-autonym="नेपाल भाषा" data-language-local-name="newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) - neerlandés" lang="nl" hreflang="nl" data-title="Pi (wiskunde)" 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/Pi" title="Pi - norvegian nynorsk" lang="nn" hreflang="nn" data-title="Pi" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegian 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/Pi" title="Pi - norvegian bokmål" lang="nb" hreflang="nb" data-title="Pi" data-language-autonym="Norsk bokmål" data-language-local-name="norvegian bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AA%E0%AC%BE%E0%AC%87" title="ପାଇ - oriya" lang="or" hreflang="or" data-title="ପାଇ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="oriya" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи - osseta" lang="os" hreflang="os" data-title="Пи" data-language-autonym="Ирон" data-language-local-name="osseta" 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%AA%E0%A8%BE%E0%A8%88" title="ਪਾਈ - punjabi" lang="pa" hreflang="pa" data-title="ਪਾਈ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Pi_(nombe)" title="Pi (nombe) - picard" lang="pcd" hreflang="pcd" data-title="Pi (nombe)" data-language-autonym="Picard" data-language-local-name="picard" class="interlanguage-link-target"><span>Picard</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl - alemany palatí" lang="pfl" hreflang="pfl" data-title="Kreiszahl" data-language-autonym="Pälzisch" data-language-local-name="alemany palatí" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pi" title="Pi - polonés" lang="pl" hreflang="pl" data-title="Pi" data-language-autonym="Polski" data-language-local-name="polonés" 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_%C3%ABd_Ludolph" title="Nùmer ëd Ludolph - piemontès" lang="pms" hreflang="pms" data-title="Nùmer ëd Ludolph" data-language-autonym="Piemontèis" data-language-local-name="piemontès" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%BE%D8%A7%D8%A6%DB%8C" title="پائی - Western Punjabi" lang="pnb" hreflang="pnb" data-title="پائی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%BE%D8%A7%DB%8C_(_%D9%81%D8%B2%D9%8A%DA%A9_)" title="پای ( فزيک ) - pashto" lang="ps" hreflang="ps" data-title="پای ( فزيک )" data-language-autonym="پښتو" data-language-local-name="pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Pi" title="Pi - portugués" lang="pt" hreflang="pt" data-title="Pi" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chiqaluwa" title="Chiqaluwa - quechua" lang="qu" hreflang="qu" data-title="Chiqaluwa" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://ro.wikipedia.org/wiki/Pi" title="Pi - romanés" lang="ro" hreflang="ro" data-title="Pi" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-roa-tara mw-list-item"><a href="https://roa-tara.wikipedia.org/wiki/Pi_greche" title="Pi greche - Tarantino" lang="nap-x-tara" hreflang="nap-x-tara" data-title="Pi greche" data-language-autonym="Tarandíne" data-language-local-name="Tarantino" class="interlanguage-link-target"><span>Tarandíne</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="Пи (число) - rus" lang="ru" hreflang="ru" data-title="Пи (число)" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A7%D1%96%D1%81%D0%BB%D0%BE_%D0%BF%D1%96" title="Чісло пі - ruthène" lang="rue" hreflang="rue" data-title="Чісло пі" data-language-autonym="Русиньскый" data-language-local-name="ruthène" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्या - sanscrit" lang="sa" hreflang="sa" data-title="प्या" data-language-autonym="संस्कृतम्" data-language-local-name="sanscrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи - iacut" lang="sah" hreflang="sah" data-title="Пи" data-language-autonym="Саха тыла" data-language-local-name="iacut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AF%E1%B1%9F%E1%B1%AD" title="ᱯᱟᱭ - santali" lang="sat" hreflang="sat" data-title="ᱯᱟᱭ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Pi_grecu" title="Pi grecu - sard" lang="sc" hreflang="sc" data-title="Pi grecu" data-language-autonym="Sardu" data-language-local-name="sard" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Pi_grecu" title="Pi grecu - sicilian" lang="scn" hreflang="scn" data-title="Pi grecu" data-language-autonym="Sicilianu" data-language-local-name="sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Pi" title="Pi - escossés" lang="sco" hreflang="sco" data-title="Pi" data-language-autonym="Scots" data-language-local-name="escossés" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Pi" title="Pi - serbocroat" lang="sh" hreflang="sh" data-title="Pi" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Pi" title="Pi - taixelhit" lang="shi" hreflang="shi" data-title="Pi" data-language-autonym="Taclḥit" data-language-local-name="taixelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B6%BA%E0%B7%92_(%E0%B6%85%E0%B6%82%E0%B6%9A%E0%B6%BA)" title="පයි (අංකය) - singalès" lang="si" hreflang="si" data-title="පයි (අංකය)" data-language-autonym="සිංහල" data-language-local-name="singalès" 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/Pi" title="Pi - Simple English" lang="en-simple" hreflang="en-simple" data-title="Pi" 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/Ludolfovo_%C4%8D%C3%ADslo" title="Ludolfovo číslo - eslovac" lang="sk" hreflang="sk" data-title="Ludolfovo číslo" data-language-autonym="Slovenčina" data-language-local-name="eslovac" 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/Pi" title="Pi - eslovèn" lang="sl" hreflang="sl" data-title="Pi" data-language-autonym="Slovenščina" data-language-local-name="eslovèn" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Summad_(Pi)" title="Summad (Pi) - somali" lang="so" hreflang="so" data-title="Summad (Pi)" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri_pi" title="Numri pi - albanés" lang="sq" hreflang="sq" data-title="Numri pi" data-language-autonym="Shqip" data-language-local-name="albanés" 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%9F%D0%B8" title="Пи - serbi" lang="sr" hreflang="sr" data-title="Пи" data-language-autonym="Српски / srpski" data-language-local-name="serbi" 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/Pi" title="Pi - suedés" lang="sv" hreflang="sv" data-title="Pi" data-language-autonym="Svenska" data-language-local-name="suedés" 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/Pai" title="Pai - swahili" lang="sw" hreflang="sw" data-title="Pai" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Pi" title="Pi - silesià" lang="szl" hreflang="szl" data-title="Pi" data-language-autonym="Ślůnski" data-language-local-name="silesià" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%88_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AE%BF%E0%AE%B2%E0%AE%BF)" title="பை (கணித மாறிலி) - tamol" lang="ta" hreflang="ta" data-title="பை (கணித மாறிலி)" data-language-autonym="தமிழ்" data-language-local-name="tamol" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B1%88" title="పై - telogó" lang="te" hreflang="te" data-title="పై" data-language-autonym="తెలుగు" data-language-local-name="telogó" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9F%D3%A3_(%D0%B0%D0%B4%D0%B0%D0%B4)" title="Пӣ (адад) - tajic" lang="tg" hreflang="tg" data-title="Пӣ (адад)" data-language-autonym="Тоҷикӣ" data-language-local-name="tajic" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B2%E0%B8%A2_(%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%84%E0%B8%87%E0%B8%95%E0%B8%B1%E0%B8%A7)" title="พาย (ค่าคงตัว) - tai" lang="th" hreflang="th" data-title="พาย (ค่าคงตัว)" data-language-autonym="ไทย" data-language-local-name="tai" 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/Pi" title="Pi - tagal" lang="tl" hreflang="tl" data-title="Pi" data-language-autonym="Tagalog" data-language-local-name="tagal" 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/Pi_say%C4%B1s%C4%B1" title="Pi sayısı - turc" lang="tr" hreflang="tr" data-title="Pi sayısı" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9F%D0%B8_%D1%81%D0%B0%D0%BD%D1%8B" title="Пи саны - tatar" lang="tt" hreflang="tt" data-title="Пи саны" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BF%D1%96" title="Число пі - ucrainés" lang="uk" hreflang="uk" data-title="Число пі" data-language-autonym="Українська" data-language-local-name="ucrainés" 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/%D9%BE%D8%A7%D8%A6%DB%8C" title="پائی - ordó" lang="ur" hreflang="ur" data-title="پائی" data-language-autonym="اردو" data-language-local-name="ordó" 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/Pi" title="Pi - ozbèc" lang="uz" hreflang="uz" data-title="Pi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ozbèc" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Pi_greco" title="Pi greco - vènet" lang="vec" hreflang="vec" data-title="Pi greco" data-language-autonym="Vèneto" data-language-local-name="vènet" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Pi_(lugu)" title="Pi (lugu) - vepse" lang="vep" hreflang="vep" data-title="Pi (lugu)" data-language-autonym="Vepsän kel’" data-language-local-name="vepse" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="articles de qualitat"><a href="https://vi.wikipedia.org/wiki/Pi" title="Pi - vietnamian" lang="vi" hreflang="vi" data-title="Pi" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamian" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pi" title="Pi - waray" lang="war" hreflang="war" data-title="Pi" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 - xinès wu" lang="wuu" hreflang="wuu" data-title="圓周率" data-language-autonym="吴语" data-language-local-name="xinès 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%9F%D0%B8" title="Пи - kalmoc" lang="xal" hreflang="xal" data-title="Пи" data-language-autonym="Хальмг" data-language-local-name="kalmoc" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Phi" title="Phi - xhòsa" lang="xh" hreflang="xh" data-title="Phi" data-language-autonym="IsiXhosa" data-language-local-name="xhòsa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%99" title="פי - yiddish" lang="yi" hreflang="yi" data-title="פי" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Pi" title="Pi - yoruba" lang="yo" hreflang="yo" data-title="Pi" data-language-autonym="Yorùbá" data-language-local-name="yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) - zelandès" lang="zea" hreflang="zea" data-title="Pi (wiskunde)" data-language-autonym="Zeêuws" data-language-local-name="zelandès" class="interlanguage-link-target"><span>Zeêuws</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="bons articles"><a href="https://zh.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 - chinés" lang="zh" hreflang="zh" data-title="圓周率" data-language-autonym="中文" data-language-local-name="chinés" 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%9C%93%E5%91%A8%E7%8E%87" title="圓周率 - xinès clàssic" lang="lzh" hreflang="lzh" data-title="圓周率" data-language-autonym="文言" data-language-local-name="xinès clàssic" 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/%C3%8E%E2%81%BF-chiu-lu%CC%8Dt" title="Îⁿ-chiu-lu̍t - xinès min del sud" lang="nan" hreflang="nan" data-title="Îⁿ-chiu-lu̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" 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%9C%93%E5%91%A8%E7%8E%87" title="圓周率 - cantonés" lang="yue" hreflang="yue" data-title="圓周率" 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/Q167#sitelinks-wikipedia" title="Modificar los ligams interlenga" class="wbc-editpage">Modificar los ligams</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="Espacis de noms"> <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/Pi" title="Veire l’article [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discutir:Pi&action=edit&redlink=1" rel="discussion" class="new" title="Discussion a prepaus d'aquesta pagina (la pagina existís pas) [t]" accesskey="t"><span>Discussion</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="Canvia la variant de llengua" > <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">occitan</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="Afichatges"> <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/Pi"><span>Legir</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pi&veaction=edit" title="Modificar aquela pagina [v]" accesskey="v"><span>Modificar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pi&action=edit" title="Modificar lo còdi font d'aquela pagina [e]" accesskey="e"><span>Modificar lo còdi</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pi&action=history" title="Los autors e versions precedentas d'aquesta pagina. [h]" accesskey="h"><span>Veire l'istoric</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <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="Bóstia d'aisinas" > <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">Bóstia d'aisinas</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">Aisinas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">Escondre</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mai d'opcions" > <div class="vector-menu-heading"> Accions </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/Pi"><span>Legir</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Pi&veaction=edit" title="Modificar aquela pagina [v]" accesskey="v"><span>Modificar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Pi&action=edit" title="Modificar lo còdi font d'aquela pagina [e]" accesskey="e"><span>Modificar lo còdi</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Pi&action=history"><span>Veire l'istoric</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generau </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Paginas_ligadas/Pi" title="Lista de las paginas ligadas a aquesta. [j]" accesskey="j"><span>Paginas connèxas</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguit_dels_ligams/Pi" rel="nofollow" title="Lista dels darrièrs cambiaments de las paginas ligadas a aquesta [k]" accesskey="k"><span>Seguit dels ligams</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=oc" title="Mandar un imatge o fichièr mèdia sul servidor [u]" accesskey="u"><span>Importar un fichièr</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:Paginas_especialas" title="Lista de totas las paginas especialas [q]" accesskey="q"><span>Paginas especialas</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Pi&oldid=2438332" title="Ligam permanent cap a aquesta version de la pagina"><span>Ligam permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Pi&action=info" title="Mai d’informacion sus aquesta pagina"><span>Informacion sus la pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Pi&id=2438332&wpFormIdentifier=titleform" title="Informacions sus cossí citar aquesta pagina"><span>Citar aqueste article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Foc.wikipedia.org%2Fwiki%2FPi"><span>Obténer una URL acorchida</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Foc.wikipedia.org%2Fwiki%2FPi"><span>Telecargar lo còdi 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"> Imprimir / exportar </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:Libre&bookcmd=book_creator&referer=Pi"><span>Crear un libre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Pi&action=show-download-screen"><span>Telecargar coma PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Pi&printable=yes" title="Version imprimibla d'aquesta pagina [p]" accesskey="p"><span>Version imprimibla</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"> Dins d'autres projèctes </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:Pi" 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/Q167" title="Ligam cap a l’element de referencial de las donadas connectadas [g]" accesskey="g"><span>Element 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="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <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">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">Escondre</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-1000_articles_fondamentals" class="mw-indicator"><div class="mw-parser-output"><span title="Tièra de 1000 articles que totas las Wikipèdias deurián aver."><span class="mw-default-size mw-image-border" typeof="mw:File"><a href="//oc.wikipedia.org/wiki/Wikip%C3%A8dia:Ti%C3%A8ra_de_1000_articles_que_totas_las_Wikip%C3%A8dias_deuri%C3%A1n_aver"><img alt="Tièra de 1000 articles que totas las Wikipèdias deurián aver." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/1000F.png/24px-1000F.png" decoding="async" width="24" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/1000F.png/36px-1000F.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/1000F.png/49px-1000F.png 2x" data-file-width="1001" data-file-height="908" /></a></span></span></div></div> </div> <div id="siteSub" class="noprint">Un article de Wikipèdia, l'enciclopèdia liura.</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="oc" dir="ltr"><p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Pi-unrolled-720.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/310px-Pi-unrolled-720.gif" decoding="async" width="310" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/465px-Pi-unrolled-720.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/620px-Pi-unrolled-720.gif 2x" data-file-width="720" data-file-height="228" /></a><figcaption>Se lo diamètre dau cercle vau 1, sa circonferéncia vau π.</figcaption></figure> <p>Lo <a href="/wiki/Nombre" title="Nombre">nombre</a> <b>pi</b>, notat amb la <a href="/wiki/Alfabet_gr%C3%A8c" title="Alfabet grèc">letra grèga</a> dau meteis nom <b><a href="/wiki/Pi_(letra_gr%C3%A8ga)" title="Pi (letra grèga)">π</a></b>, es una <a href="/wiki/Constanta_matematica" title="Constanta matematica">constanta matematica</a> que sa valor es lo repòrt entre la <a href="/wiki/Circonfer%C3%A9ncia" title="Circonferéncia">circonferéncia</a> d’un <a href="/wiki/Cercle" title="Cercle">cercle</a> quin que siá e son <a href="/wiki/Diam%C3%A8tre" title="Diamètre">diamètre</a>, en <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">geometria euclidiana</a>; es tanben la valor dau repòrt entre la <a href="/wiki/Aira" title="Aira">superfícia</a> d'un cercle e lo <a href="/w/index.php?title=Carrat_(alg%C3%A8bra)&action=edit&redlink=1" class="new" title="Carrat (algèbra) (la pagina existís pas)">carrat</a> de son <a href="/wiki/Rai_(geometria)" title="Rai (geometria)">rai</a>. </p><p>Sonat tanben <span class="citation">«constanta d'Arquimèdes»</span>, lo nombre π a per valor aproximada, en <a href="/w/index.php?title=Desvolopament_decimau&action=edit&redlink=1" class="new" title="Desvolopament decimau (la pagina existís pas)">escritura decimala</a>, 3,141593. De formulas scientificas nombrosas, dins de domenis coma la <a href="/wiki/Fisica" title="Fisica">fisica</a>, l'<a href="/wiki/Engenhari%C3%A1" title="Engenhariá">engenhariá</a> e de segur lei <a href="/wiki/Matematicas" title="Matematicas">matematicas</a>, fan intervenir π, qu'es una dei constantas matematicas mai importantas<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. </p><p>Aqueu nombre es <a href="/w/index.php?title=Nombre_irracionau&action=edit&redlink=1" class="new" title="Nombre irracionau (la pagina existís pas)">irracionau</a>: autrament dich, se pòt pas exprimir coma lo quocient de dos <a href="/wiki/Nombre_entier" class="mw-redirect" title="Nombre entier">nombres entiers</a>; aiçò implica que son escritura decimala es ni finida, ni periodica. Es tanben <a href="/w/index.php?title=Nombre_transcendent&action=edit&redlink=1" class="new" title="Nombre transcendent (la pagina existís pas)">transcendent</a>, çò que vòu dire qu’existís pas de <a href="/w/index.php?title=Polin%C3%B2mi&action=edit&redlink=1" class="new" title="Polinòmi (la pagina existís pas)">polinòmi</a> non nul de coeficients entiers que π ne siá una <a href="/w/index.php?title=Z%C3%A8ro_d%27una_foncion&action=edit&redlink=1" class="new" title="Zèro d'una foncion (la pagina existís pas)">raiç</a>; se deu a <a href="/w/index.php?title=Ferdinand_von_Lindemann&action=edit&redlink=1" class="new" title="Ferdinand von Lindemann (la pagina existís pas)">Ferdinand von Lindemann</a> la demostracion d'aqueu resultat en 1882. La determinacion de valors aproximadas pron precisas de π e la comprension de sa natura son de questions qu'an traversat l'<a href="/w/index.php?title=Ist%C3%B2ria_dei_matematicas&action=edit&redlink=1" class="new" title="Istòria dei matematicas (la pagina existís pas)">istòria dei matematicas</a>. </p><p>La letra π que nòta aqueu nombre es simplament l'iniciala dau mot <a href="/wiki/Gr%C3%A8c_(lenga)" title="Grèc (lenga)">grèc</a> περίμετρος, <span class="citation">«perimètre»</span>. Foguèt utilizada premier per William Jones en 1707 puei popularizada per <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> en 1737<sup id="cite_ref-collier_2-0" class="reference"><a href="#cite_note-collier-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicions_e_premierei_proprietats">Definicions e premierei proprietats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=1" title="Modificar la seccion : Definicions e premierei proprietats" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=1" title="Edita el codi de la secció: Definicions e premierei proprietats"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-section bandeau-niveau-detail loupe">Article detalhat: <a href="/w/index.php?title=Ist%C3%B2ria_de_Pi&action=edit&redlink=1" class="new" title="Istòria de Pi (la pagina existís pas)">Istòria de Pi</a>.</div> <div class="mw-heading mw-heading3"><h3 id="Definicions">Definicions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=2" title="Modificar la seccion : Definicions" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=2" title="Edita el codi de la secció: Definicions"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Catalan-Pi_deffinition.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Catalan-Pi_deffinition.PNG/220px-Catalan-Pi_deffinition.PNG" decoding="async" width="220" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/75/Catalan-Pi_deffinition.PNG 1.5x" data-file-width="287" data-file-height="320" /></a><figcaption>Circonferéncia = π × diamètre</figcaption></figure> <p>En <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">geometria euclidiana</a>, π se definís coma lo repòrt entre la <a href="/wiki/Circonfer%C3%A9ncia" title="Circonferéncia">circonferéncia</a> <i>C</i> d'un <a href="/wiki/Cercle" title="Cercle">cercle</a> e son <a href="/wiki/Diam%C3%A8tre" title="Diamètre">diamètre</a> <i>d</i><sup id="cite_ref-adm_3-0" class="reference"><a href="#cite_note-adm-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {C}{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {C}{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98a23e73a342246e95838018afd6f157a859564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.68ex; height:5.509ex;" alt="{\displaystyle \pi ={\frac {C}{d}}.}"></span></dd></dl> <p>Lo repòrt <sup><i>C</i></sup>/<sub><i>d</i></sub> es constant; autrament dich, depend pas de la talha dau cercle. Per exemple, se de dos cercles, lo diamètre dau premier es doble d'aqueu dau segond, la circonferéncia dau premier es tanben dobla d'aquela dau segond: lo repòrt <sup><i>C</i></sup>/<sub><i>d</i></sub> càmbia pas. </p><p>Se pòt tanben definir π coma lo repòrt entre la <a href="/wiki/Aira" title="Aira">superfícia</a> <i>A</i> d'un cercle e la superfícia d'un carrat qu'a per costat lo <a href="/wiki/Rai_(geometria)" title="Rai (geometria)">rai</a> dau cercle<sup id="cite_ref-adm_3-1" class="reference"><a href="#cite_note-adm-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {A}{r^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {A}{r^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e2b4d298896dd3662e9e1421fda20471a10d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.016ex; height:5.676ex;" alt="{\displaystyle \pi ={\frac {A}{r^{2}}}.}"></span></dd></dl> <p>Aquelei definicions se justifican per cèrtei proprietats de la geometria euclidiana, coma la qu'enóncia que totei lei cercles son <a href="/w/index.php?title=Similitud_(geometria)&action=edit&redlink=1" class="new" title="Similitud (geometria) (la pagina existís pas)">semblables</a>. Mai lo nombre π apareis dins de domenis dei matematicas que son pas directament liats a la geometria. Dins leis expausats axiomatics dei matematicas, sovent se definís π a partir de l'<a href="/w/index.php?title=Analisi_(matematicas)&action=edit&redlink=1" class="new" title="Analisi (matematicas) (la pagina existís pas)">analisi matematica</a>: se pòt donar una definicion dei <a href="/w/index.php?title=Foncion_trigonometrica&action=edit&redlink=1" class="new" title="Foncion trigonometrica (la pagina existís pas)">foncions trigonometricas</a> (cos, sin) independenta de la geometria dau cercle, puei definir π coma lo doble dau pus pichon nombre positiu <i>x</i> tau que cos(<i>x</i>) = 0<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>. Lei formulas balhadas <i>infra</i> pòdon servir de definicions equivalentas de π (pron que se justifique que definisson lo meteis nombre). </p> <div class="mw-heading mw-heading3"><h3 id="Irracionalitat">Irracionalitat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=3" title="Modificar la seccion : Irracionalitat" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=3" title="Edita el codi de la secció: Irracionalitat"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo nombre π es <a href="/w/index.php?title=Nombre_irracionau&action=edit&redlink=1" class="new" title="Nombre irracionau (la pagina existís pas)">irracionau</a>, çò que significa qu'es impossible de trobar d'<a href="/wiki/Nombre_entier" class="mw-redirect" title="Nombre entier">entiers</a> <i>p</i> e <i>q</i> taus que <i>π=p/q</i>. <a href="/wiki/Muhammad_ibn_Musa_al-Khwarizmi" title="Muhammad ibn Musa al-Khwarizmi">Al-Khawarizmi</a> conjectura tre lo <a href="/wiki/S%C3%A8gle_IX" title="Sègle IX">sègle IX</a> que π es irracionau<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>. <a href="/wiki/Maimonides" title="Maimonides">Moshé Maimonides</a> menciona pereu aquela idèa au <a href="/wiki/S%C3%A8gle_XII" title="Sègle XII">sègle XII</a>. Pasmens faudrà esperar lo <a href="/wiki/S%C3%A8gle_XVIII" title="Sègle XVIII">sègle XVIII</a> per que <a href="/w/index.php?title=Johann_Heinrich_Lambert&action=edit&redlink=1" class="new" title="Johann Heinrich Lambert (la pagina existís pas)">Johann Heinrich Lambert</a> ne fague la demostracion<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>. </p><p>Es en 1761 que dins son <i>Mémoires sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques</i>, Lambert estúdia lo desvolopament en fraccion continua de tan(<i>x</i>) e demòstra que lo desvolopament en fraccion continua de tan(<i>m</i>/<i>n</i>) (ont <i>m</i>, <i>n</i> son d'entiers naturaus diferents de 0) es: </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 \tan \left({\frac {m}{n}}\right)={\frac {m\mid }{\mid n}}-{\frac {m^{2}\mid }{\mid 3n}}-{\frac {m^{2}\mid }{\mid 5n}}-{\frac {m^{2}\mid }{\mid 7n}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">∣</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">∣</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> <mn>3</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">∣</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> <mn>5</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">∣</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> <mn>7</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {m}{n}}\right)={\frac {m\mid }{\mid n}}-{\frac {m^{2}\mid }{\mid 3n}}-{\frac {m^{2}\mid }{\mid 5n}}-{\frac {m^{2}\mid }{\mid 7n}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4eaca66a45cdd1af08f1bda1cb45c6dfeef7c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.451ex; height:6.676ex;" alt="{\displaystyle \tan \left({\frac {m}{n}}\right)={\frac {m\mid }{\mid n}}-{\frac {m^{2}\mid }{\mid 3n}}-{\frac {m^{2}\mid }{\mid 5n}}-{\frac {m^{2}\mid }{\mid 7n}}+\cdots }"></span>.</dd></dl> <p>Per consequent, quand <i>x</i> es diferent de 0 e racionau, lo desvolopament en fraccion continua de tan(<i>x</i>) es illimitat; mai se sap qu’un tau desvolopament illimitat mena a un nombre irracionau. Fin finala, quand <i>x</i> es diferent de 0 e racionau, tan(<i>x</i>) es irracionau.<br /> Se'n dedutz l'irracionalitat de π: se lo nombre π foguèsse racionau, tanben o seriá π/4, doncas tan(π/4) seriá irracionau; coma tan(π/4) = 1, qu'es racionau, es absurde. </p><p>Au <a href="/wiki/S%C3%A8gle_XX" title="Sègle XX">sègle XX</a>, s'es trobat d'autrei demostracions d'irracionalitat qu'exigisson pas de conoissenças pus avançadas que leis elements dau calcul integrau. Una d'entre elei, que se deu a Ivan Niven, es ben coneguda<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><sup class="reference cite_virgule">,</sup><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>. Mary Cartwright aviá trobat un pauc aperavans una demostracion analòga<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>. </p> <div class="mw-heading mw-heading3"><h3 id="Transcendéncia"><span id="Transcend.C3.A9ncia"></span>Transcendéncia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=4" title="Modificar la seccion : Transcendéncia" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=4" title="Edita el codi de la secció: Transcendéncia"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo nombre π es tanben <a href="/w/index.php?title=Nombre_transcendent&action=edit&redlink=1" class="new" title="Nombre transcendent (la pagina existís pas)">transcendent</a>, valent a dire qu'existís ges de <a href="/w/index.php?title=Polin%C3%B2mi&action=edit&redlink=1" class="new" title="Polinòmi (la pagina existís pas)">polinòmi</a> de coeficients racionaus admetent π per <a href="/w/index.php?title=Z%C3%A9ro_d%27une_foncion&action=edit&redlink=1" class="new" title="Zéro d'une foncion (la pagina existís pas)">raiç</a><sup id="cite_ref-ttop_11-0" class="reference"><a href="#cite_note-ttop-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>. </p><p>Es au <a href="/wiki/S%C3%A8gle_XIX" title="Sègle XIX">sègle XIX</a> que se demostrèt aqueu resultat. En 1873, <a href="/w/index.php?title=Charles_Hermite&action=edit&redlink=1" class="new" title="Charles Hermite (la pagina existís pas)">Charles Hermite</a> provèt que la basa dau <a href="/wiki/Logaritme_neperian" title="Logaritme neperian">logaritme neperian</a>, lo <a href="/wiki/E_(nombre)" title="E (nombre)">nombre e</a>, es transcendenta. En 1882, <a href="/w/index.php?title=Ferdinand_von_Lindemann&action=edit&redlink=1" class="new" title="Ferdinand von Lindemann (la pagina existís pas)">Ferdinand von Lindemann</a> generalizèt son metòde en un teorèma (<a href="/w/index.php?title=Teor%C3%A8ma_d%27Hermite-Lindemann&action=edit&redlink=1" class="new" title="Teorèma d'Hermite-Lindemann (la pagina existís pas)">Teorèma d'Hermite-Lindemann</a>) qu'estipula que, se <i>x</i> es un nombre reau o complèxe diferent de 0 e <a href="/wiki/Nombre_algebric" title="Nombre algebric">algebric</a> (çò es non transcendent), alora e<sup><i>x</i></sup> es transcendent.<br /> Se'n dedutz la transcendéncia de π: se lo nombre π foguèsse algebric, tanben o seriá iπ, doncas e<sup>iπ</sup> seriá transcendent; coma e<sup>iπ</sup> = cos(π) + i sin(π) = -1, qu'es algebric, es absurde. </p><p>Una consequéncia importanta de la transcendéncia de π es qu'aquest nombre es pas <a href="/w/index.php?title=Nombre_constructible&action=edit&redlink=1" class="new" title="Nombre constructible (la pagina existís pas)">constructible</a>. D'efècte, lo <a href="/w/index.php?title=Teor%C3%A8ma_de_Wantzel&action=edit&redlink=1" class="new" title="Teorèma de Wantzel (la pagina existís pas)">teorèma de Wantzel</a> enóncia entre autrei que tot nombre constructible es algebric. Dau fach que lei coordenadas de totei lei ponchs que se pòdon construire amb la règla e lo compàs son de nombres constructibles, la <a href="/w/index.php?title=Q%C3%BCadratura_dau_cercle&action=edit&redlink=1" class="new" title="Qüadratura dau cercle (la pagina existís pas)">qüadratura dau cercle</a> es impossibla: se pòt pas construire, solament amb la règla e lo compàs, un carrat que sa superfícia seriá la d'un cercle donat<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>. Aqueu resultat es important istoricament, car la qüadratura dau cercle es un dei problèmas celèbres de geometria elementària que nos leguèron lei matematicians grècs de l'Antiquitat. </p> <div class="mw-heading mw-heading3"><h3 id="Representacion_decimala">Representacion decimala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=5" title="Modificar la seccion : Representacion decimala" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=5" title="Edita el codi de la secció: Representacion decimala"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lei 50 premierei chifras de l'escritura decimala de π son: </p> <dl><dd>3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510</dd> <dd><i>Vejatz <a href="#Liames_extèrnes">lei liames extèrnes</a> per mai de decimalas.</i></dd></dl> <p>Mentre qu'a l'ora d'ara, se conois mai de 10<sup>12</sup> decimalas de π<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>, dins leis aplicacions s'utiliza fòrça rarament mai d'un desenau de chifras. Per exemple, l'aproximacion de π que s'obtèn en conservant lei 39 premierei decimalas sufís per calcular la circonferéncia d'un cercle que sei dimensions son aquelei de l'univèrs observable amb una precision de l'òrdre dau rai d'un atòm d'<a href="/wiki/Idrog%C3%A8n" title="Idrogèn">idrogèn</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>. </p><p>Estent que π es irracionau, sa representacion decimala es illimitada e non periodica. La seguida dei decimalas de π a totjorn pivelat matematicians e amators, e s'es consacrat abòrd d'esfòrç dins lei darriers sègles per obtenir de mai en mai de decimalas e n'estudiar lei proprietats<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>. Se conois ara mai de mila miliards de decimalas de π. Maugrat lei recèrcas efectuadas, se i es decelat ges d'estructura, e lo nombre π sembla de se comportar coma un generador de nombres aleatòris<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>. Lei chifras de la representacion decimala de π son disponiblas subre fòrça paginas d'Internet; existis de logiciaus de calcul dei decimalas de π que ne pòdon determinar de miliards e que se pòt installar sus un ordinator personau quin que siá. </p><p>D'autra part, lo desvolopament decimau de π duèrbe d’autrei questions, en particular de saber se π es un <a href="/w/index.php?title=Nombre_normau&action=edit&redlink=1" class="new" title="Nombre normau (la pagina existís pas)">nombre normau</a>, valent a dire se sei chifras en escritura decimala son equirepartidas. Òm se pòt tanben demandar se π es un <a href="/w/index.php?title=Nombre_univ%C3%A8rs&action=edit&redlink=1" class="new" title="Nombre univèrs (la pagina existís pas)">nombre univèrs</a>, çò es se se pòt trobar dins son desvolopament decimau una sequéncia predefinida de chifras, quina que siá (per exemple: 123 456 789 987 654 320). Uei, se conois pas de respònsa a aquelei questions<sup id="cite_ref-Conférence_Delahaye_18-0" class="reference"><a href="#cite_note-Conférence_Delahaye-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Aproximacion_de_π"><span id="Aproximacion_de_.CF.80"></span>Aproximacion de π</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=6" title="Modificar la seccion : Aproximacion de π" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=6" title="Edita el codi de la secció: Aproximacion de π"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se pòt trobar empiricament una valor aproximada de π, per mejan d'un grand cercle que se'n mesura lo diamètre e la circonferéncia, puis en devesissent la circonferéncia per lo diamètre. Una autre encaminament geometric, atribuit a <a href="/wiki/Arquim%C3%A8des" title="Arquimèdes">Arquimèdes</a><sup id="cite_ref-NOVA_19-0" class="reference"><a href="#cite_note-NOVA-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>, consistís a calcular lo <a href="/wiki/Perim%C3%A8tre" title="Perimètre">perimètre</a> <i>P<sub>n</sub></i> d'un <a href="/w/index.php?title=Polig%C3%B2n_regular&action=edit&redlink=1" class="new" title="Poligòn regular (la pagina existís pas)">poligòn regular</a> de <i>n</i> costats <a href="/w/index.php?title=Cercle_circonscrich&action=edit&redlink=1" class="new" title="Cercle circonscrich (la pagina existís pas)">circonscrich</a> a un cercle de diamètre <i>d</i>. S'obtèn alora π gràcias a la formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{n\to \infty }{\frac {P_{n}}{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{n\to \infty }{\frac {P_{n}}{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef45806eea40b08ab7d37374f45db54673d2508a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.284ex; height:5.343ex;" alt="{\displaystyle \pi =\lim _{n\to \infty }{\frac {P_{n}}{d}}.}"></span></dd></dl> <p>Au mai lo nombre <i>n</i> de costats dau poligòn es grand, au mai es precisa l'aproximacion de π per lo quocient <i>P<sub>n</sub></i> / <i>d</i>. Arquimèdes determinèt la precision dau metòde en comparant lo perimètre dau poligòn circonscrich amb aqueu d'un poligòn regular dau meteis nombre de costats <a href="/w/index.php?title=Cercle_inscrich&action=edit&redlink=1" class="new" title="Cercle inscrich (la pagina existís pas)">inscrich</a> dins lo cercle. Amb dos poligòns de 96 costats, establiguèt que: </p> <dl><dd>3 + 10/71 < π < 3 + 1/7<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>.</dd></dl> <p>Tanben se pòt obtenir de valors aproximadas de π per mejan de metòdes d'analisi matematica. La màger part dei formulas que s'utiliza per calcular π se fondan subre sei proprietats analiticas e son de mau comprendre sensa conoissenças en <a href="/wiki/Trigonometria" title="Trigonometria">trigonometria</a> e <a href="/wiki/Calcul_integrau" class="mw-redirect" title="Calcul integrau">calcul integrau</a>. Pasmens, d'unei son pron simplas, coma la formula de Leibniz<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}\cdots \right).\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}\cdots \right).\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3483267fb09292e569b362869d88e9e8149b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.204ex; width:56.321ex; height:7.176ex;" alt="{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}\cdots \right).\!}"></span></dd></dl> <p>Mai aquela seria convergís tròp lentament e, maugrat sa simplicitat aparenta, pòt pas servir tala e quala per calcular d'aproximacions de π (fau gaireben 300 tèrmes per obtenir doas decimalas exactas)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>. Pasmens, es possible d'accelerar la convergéncia: se definís a partir de la precedenta una autra seguida que convergís vèrs π fòrça mai rapidament, en pausant: </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 p_{0,1}={\frac {1}{1}},\ p_{0,2}={\frac {1}{1}}-{\frac {1}{3}},\ p_{0,3}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}},\ p_{0,4}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}},\cdots \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>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> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0,1}={\frac {1}{1}},\ p_{0,2}={\frac {1}{1}}-{\frac {1}{3}},\ p_{0,3}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}},\ p_{0,4}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}},\cdots \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886f7f18b849514f7098de3df78bb3cd0fe754e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; margin-right: -0.204ex; width:71.941ex; height:5.343ex;" alt="{\displaystyle p_{0,1}={\frac {1}{1}},\ p_{0,2}={\frac {1}{1}}-{\frac {1}{3}},\ p_{0,3}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}},\ p_{0,4}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}},\cdots \!}"></span></dd></dl> <p>e en definissent: </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 p_{i,j}={\frac {p_{i-1,j}+p_{i-1,j+1}}{2}}{\text{ per tot pareu }}(i,\,j){\text{ tau que }}i\geq 1{\text{ e }}j\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> per tot pareu </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>j</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> tau que </mtext> </mrow> <mi>i</mi> <mo>≥</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> e </mtext> </mrow> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i,j}={\frac {p_{i-1,j}+p_{i-1,j+1}}{2}}{\text{ per tot pareu }}(i,\,j){\text{ tau que }}i\geq 1{\text{ e }}j\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc4c8afa0bf577eb4e9208cfefc00d50871dfeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:62.821ex; height:5.343ex;" alt="{\displaystyle p_{i,j}={\frac {p_{i-1,j}+p_{i-1,j+1}}{2}}{\text{ per tot pareu }}(i,\,j){\text{ tau que }}i\geq 1{\text{ e }}j\geq 1}"></span>.</dd></dl> <p>Lo calcul 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 p_{10,10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{10,10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdca6f29a17ef05740d6970946f4029fd119884f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:5.236ex; height:2.343ex;" alt="{\displaystyle p_{10,10}}"></span> demanda aperaquí lo meteis temps qu'aqueu que fau per obtenir lei 150 premiers tèrmes de la seria iniciala, mai la precision es ben melhora: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{10,10}=4p_{10,10}=3,141592653\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>141592653</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{10,10}=4p_{10,10}=3,141592653\ldots \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa430204e760b5baeac533acaa3d105f3720ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.223ex; height:2.843ex;" alt="{\displaystyle \pi _{10,10}=4p_{10,10}=3,141592653\ldots \;}"></span> es una aproximacion qu'a 9 decimalas exactas.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Utilizacion_en_matematicas_e_en_sciéncias"><span id="Utilizacion_en_matematicas_e_en_sci.C3.A9ncias"></span>Utilizacion en matematicas e en sciéncias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=7" title="Modificar la seccion : Utilizacion en matematicas e en sciéncias" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=7" title="Edita el codi de la secció: Utilizacion en matematicas e en sciéncias"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lei formulas interessantas que contenon π son innombrablas e apareisson dins gaireben totei lei domenis dei matematicas e dei sciéncias. Una dei pus celèbras après aquelei que pertòcan la definicion geometrica de π es l’<a href="/wiki/Identitat_d%27Euler" title="Identitat d'Euler">identitat d'Euler</a>. Aquela formula es estada presentada coma la formula « mai remarcabla » per sa particularitat de far intervenir 1, 0, e, i e, de segur, π, que son entre lei nombres pus « remarcables » dei matematicas. </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 e^{i\pi }+1=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\pi }+1=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9f6702338ab012224d08d5669ab190d3de55c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.734ex; height:2.843ex;" alt="{\displaystyle e^{i\pi }+1=0\;}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Geometria">Geometria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=8" title="Modificar la seccion : Geometria" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=8" title="Edita el codi de la secció: Geometria"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo nombre π apareis dins fòrça formulas de <a href="/wiki/Geometria" title="Geometria">geometria</a> pertocant lei <a href="/wiki/Cercle" title="Cercle">cercles</a> e leis <a href="/wiki/Esf%C3%A8ra" title="Esfèra">esfèras</a> </p> <table class="wikitable"> <tbody><tr> <th>Forma geometrica </th> <th>Formula </th></tr> <tr> <td><a href="/wiki/Circonfer%C3%A9ncia" title="Circonferéncia">Circonferéncia</a> d’un cercle de <a href="/wiki/Rai_(geometria)" title="Rai (geometria)">rai</a> <i>r</i> e de <a href="/wiki/Diam%C3%A8tre" title="Diamètre">diamètre</a> <i>d</i> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=2\pi r=\pi d\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo>=</mo> <mi>π</mi> <mi>d</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\pi r=\pi d\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f11237cf908fb1da67d48efdbed819150a7e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:14.441ex; height:2.176ex;" alt="{\displaystyle C=2\pi r=\pi d\,\!}"></span> </td></tr> <tr> <td><a href="/wiki/Aira" title="Aira">Aira</a> d’un <a href="/w/index.php?title=Disque_(geometria)&action=edit&redlink=1" class="new" title="Disque (geometria) (la pagina existís pas)">disque</a> de rai <i>r</i> </td> <td><span class="mwe-math-element"><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=\pi r^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201220140d2f0f989e97c68a6ed6e56a335fed93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:8.664ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}\,\!}"></span> </td></tr> <tr> <td>Aira interiora a una <a href="/wiki/Ellipsa" title="Ellipsa">ellipsa</a> de semiaxes <i>a</i> e <i>b</i> </td> <td><span class="mwe-math-element"><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=\pi ab\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi ab\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054e5e946a7a106ca68c7ca0f3cae845621b678f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:8.788ex; height:2.176ex;" alt="{\displaystyle A=\pi ab\,\!}"></span> </td></tr> <tr> <td><a href="/wiki/Volum" title="Volum">Volum</a> d’una bola de rai <i>r</i> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi d^{3}}{6}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>6</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi d^{3}}{6}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807bf81442ff7a5e2b7cf277ec5baf9cd94d5e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:18.245ex; height:5.676ex;" alt="{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi d^{3}}{6}}\,\!}"></span> </td></tr> <tr> <td>Aira d’una <a href="/wiki/Esf%C3%A8ra" title="Esfèra">esfèra</a> de rai <i>r</i> </td> <td><span class="mwe-math-element"><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=4\pi r^{2}=\pi d^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi r^{2}=\pi d^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c6abb39a2a5e681fb64de75494e8bf6afda191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:16.529ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}=\pi d^{2}\,\!}"></span> </td></tr> <tr> <td>Volum d’un <a href="/w/index.php?title=Cilindre&action=edit&redlink=1" class="new" title="Cilindre (la pagina existís pas)">cilindre</a> d'autor <i>h</i> e de rai <i>r</i> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\pi r^{2}h\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\pi r^{2}h\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b3b36fbe719757e46222c6aee4f16cd063d469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:10.047ex; height:2.676ex;" alt="{\displaystyle V=\pi r^{2}h\,\!}"></span> </td></tr> <tr> <td>Aira extèrna d’un cilindre d'autor <i>h</i> e de rai <i>r</i> </td> <td><span class="mwe-math-element"><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=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0074d2793214ffb2b33f636caaa57a703ddcc5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:34.846ex; height:3.176ex;" alt="{\displaystyle A=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!}"></span> </td></tr> <tr> <td>Volum d’un <a href="/w/index.php?title=C%C3%B2n_(geometria)&action=edit&redlink=1" class="new" title="Còn (geometria) (la pagina existís pas)">còn</a> d'autor <i>h</i> e de rai <i>r</i> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {1}{3}}\pi r^{2}h\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}\pi r^{2}h\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13aee8f9eaacb9a61619ddae79e47ce2e14010ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:12.045ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}\pi r^{2}h\,\!}"></span> </td></tr> <tr> <td>Aira extèrna d’un còn d'autor <i>h</i> e de rai <i>r</i> </td> <td><span class="mwe-math-element"><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=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58dae10ed9e58e13b3da491c0caa93a41500f8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:44.383ex; height:3.509ex;" alt="{\displaystyle A=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!}"></span> </td></tr></tbody></table> <p>L'aira <span class="citation">«laterala»</span> d’un cilindre circonscrich a l'esfèra e de meteissa autor es egala a la de l'esfèra.<br /> Tanben se tròba π dins l'expression deis airas e volums deis iperesfèras (de mai de 3 dimensions).<br /> La mesura d’<a href="/wiki/Angle" title="Angle">angle</a> 180° (en <a href="/w/index.php?title=Gra_(angle)&action=edit&redlink=1" class="new" title="Gra (angle) (la pagina existís pas)">gras</a>) es egala a π <a href="/wiki/Radian" title="Radian">radians</a>. </p><p>En <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">geometria euclidiana</a>, la soma deis angles d’un triangle es egala a π. En <a href="/w/index.php?title=Geometria_non_euclidiana&action=edit&redlink=1" class="new" title="Geometria non euclidiana (la pagina existís pas)">geometria non euclidiana</a>, aquela soma pòt èsser superiora o inferiora a π, e lo repòrt de la circonferéncia dau cercle a son diamètre pòt tanben diferir de π. </p> <div class="mw-heading mw-heading3"><h3 id="Autrei_definicions">Autrei definicions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=9" title="Modificar la seccion : Autrei definicions" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=9" title="Edita el codi de la secció: Autrei definicions"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-section bandeau-niveau-detail loupe">Article detalhat: <a href="/w/index.php?title=Exponenciala&action=edit&redlink=1" class="new" title="Exponenciala (la pagina existís pas)">Exponenciala</a>.</div> <p>La definicion istorica e usuala dau nombre π (lo repòrt de la circonferéncia d’un cercle a son diamètre) es inadaptada per desgatjar lei proprietats dau nombre π, que sòrton largament dau quadre de la geometria. Per semblant ai foncions <a href="/w/index.php?title=Cosinus&action=edit&redlink=1" class="new" title="Cosinus (la pagina existís pas)">cosinus</a> e <a href="/w/index.php?title=Sinus&action=edit&redlink=1" class="new" title="Sinus (la pagina existís pas)">sinus</a> que se definisson intuitivament en partent dau <a href="/w/index.php?title=Cercle_trigonometric&action=edit&redlink=1" class="new" title="Cercle trigonometric (la pagina existís pas)">cercle trigonometric</a> mai rigorosament per mejan dei <a href="/w/index.php?title=Seria_de_pot%C3%A9ncias&action=edit&redlink=1" class="new" title="Seria de poténcias (la pagina existís pas)">serias de poténcias</a>, se pòt definir analiticament lo nombre π, permetent son estudi gràcias ais instruments de l’<a href="/w/index.php?title=Analisi_(matematicas)&action=edit&redlink=1" class="new" title="Analisi (matematicas) (la pagina existís pas)">analisi</a>. </p><p>Lei proprietats exp(z+w)=exp(z)exp(w) e exp(0)=1 que resultan de la definicion analitica de l’exponenciala fan que l’aplicacion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t\mapsto \exp(it)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>t</mi> <mo stretchy="false">↦</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t\mapsto \exp(it)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc825c3771575f95602d5644e536ce9c619d148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.189ex; height:2.176ex;" alt="{\displaystyle \scriptstyle t\mapsto \exp(it)}"></span> es un <a href="/w/index.php?title=Morfisme&action=edit&redlink=1" class="new" title="Morfisme (la pagina existís pas)">morfisme</a> de <a href="/wiki/Grop_(matematicas)" title="Grop (matematicas)">grops</a> <a href="/w/index.php?title=Continuitat&action=edit&redlink=1" class="new" title="Continuitat (la pagina existís pas)">continu</a> dau grop additiu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (\mathbb {R} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\mathbb {R} ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4480ce7ef58fbb6793df7a2e6411ebe07743c63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.202ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\mathbb {R} ,+)}"></span> vèrs lo grop multiplicatiu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (\mathbb {U} ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\mathbb {U} ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc99aa17a208b127bb0a2e78d3abba165b72879f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.202ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\mathbb {U} ,\times )}"></span> (onte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {U} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {U} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a74e7ef7b7ad13e43c1b23f174c3e19f55b3ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {U} }"></span> es l’ensemble dei <a href="/wiki/Nombre_compl%C3%A8xe" title="Nombre complèxe">nombres complèxes</a> que son modul vau 1). Se demòstra puei que l’ensemble dei nombres reaus <i>t</i> taus que exp(i<i>t</i>) = 1 es de la forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle a\,\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle a\,\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a330fd34b0b00a9e1da89cad0c44cd006b8eef14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.353ex; height:1.676ex;" alt="{\displaystyle \scriptstyle a\,\mathbb {Z} }"></span> onte <i>a</i> es un reau estrictament positiu.<br /> Se definís alora <span class="mwe-math-element"><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 =a/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =a/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53dd1b84bbad2bbbb57e3c8e4e811a780410e8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.985ex; height:2.843ex;" alt="{\displaystyle \pi =a/2}"></span>. Lo calcul integrau permet puei de verificar qu'aquela definicion abstracha correspònd a aquela de la geometria euclidiana. </p><p>Lo grop <a href="/w/index.php?title=Bourbaki&action=edit&redlink=1" class="new" title="Bourbaki (la pagina existís pas)">Bourbaki</a> prepausa una autra definicion fòrça vesina en demostrant qu'existís un morfisme de grop <i> f</i> continu 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 \scriptstyle (\mathbb {R} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\mathbb {R} ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4480ce7ef58fbb6793df7a2e6411ebe07743c63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.202ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\mathbb {R} ,+)}"></span> vèrs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (\mathbb {U} ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\mathbb {U} ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc99aa17a208b127bb0a2e78d3abba165b72879f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.202ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\mathbb {U} ,\times )}"></span> tau que <i>f</i>(1/4) = <i>i</i>. Demòstra qu'aqueu morfisme es periodic de periòde 1, derivable e qu’existís un reau <i>a</i> tau que, per tot reau <i>x</i>, <i>f'(x) = 2iaf(x)</i>. Definís alora π = <i>a</i>. </p><p>Lei dos metòdes precedents consistisson en realitat a <a href="/w/index.php?title=Longor_d%27un_arc&action=edit&redlink=1" class="new" title="Longor d'un arc (la pagina existís pas)">rectificar</a> lo cercle siá amb la foncion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t\mapsto e^{it},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>t</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t\mapsto e^{it},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cda0ab472a11254a0e738148a31a70122121d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.567ex; height:2.176ex;" alt="{\displaystyle \scriptstyle t\mapsto e^{it},}"></span> siá amb la foncion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t\mapsto e^{2i\pi t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>t</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>π</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t\mapsto e^{2i\pi t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077ad976728e5f6e12aa0679333cb3211fd029f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.542ex; height:2.009ex;" alt="{\displaystyle \scriptstyle t\mapsto e^{2i\pi t}}"></span> </p><p>Mai se pòt tanben definir π gràcias au calcul integrau en pausant: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\int _{0}^{1}{\sqrt {1-x^{2}}}\ \mathrm {d} x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>4</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\int _{0}^{1}{\sqrt {1-x^{2}}}\ \mathrm {d} x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683b0fe8e0c7f63500834940d90dbbdf16cda91b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.186ex; height:6.176ex;" alt="{\displaystyle \pi =4\int _{0}^{1}{\sqrt {1-x^{2}}}\ \mathrm {d} x\,}"></span></dd></dl> <p>çò qu'equivau a calcular l’aira d’un quart de disc. </p><p>O ben per mejan dau denombrament, en sonant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \varphi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2d970f45ef3450ec2a367c41a48c7bd1450658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.34ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \varphi (n)}"></span> lo nombre de pareus d’entiers naturaus (<i>p</i>, <i>q</i>) taus 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 \scriptstyle p^{2}+q^{2}\leq n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle p^{2}+q^{2}\leq n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ea3f5e561aa43468d739cca051230789e5796e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.063ex; width:7.691ex; height:2.176ex;" alt="{\displaystyle \scriptstyle p^{2}+q^{2}\leq n^{2}}"></span> e en definissent: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\lim _{n\to +\infty }{\frac {\varphi (n)}{n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>4</mn> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\lim _{n\to +\infty }{\frac {\varphi (n)}{n^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5be106c043532a0a9b4d26f0a9b9e72dc5ec64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.478ex; height:6.009ex;" alt="{\displaystyle \pi =4\lim _{n\to +\infty }{\frac {\varphi (n)}{n^{2}}}}"></span></dd></dl> <p>çò qu'es un autre biais de carrar lo quart de disc. </p><p>O encara, se la foncion cosinus es estada definida analiticament (per sa seria de poténcias o per l’unica solucion <i>f</i> de l’eqüacion diferenciala <span class="mwe-math-element"><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''=-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''=-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc7293b16d52f2a42cf102e32153c63e341ef7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle y''=-y}"></span> verificant lei condicions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00884d9c3ef0dbc18fe185a4444f1da931c8ba0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=1}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,'(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,'(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76e740dad743f7856998b3c713fd3fdd8f25410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.583ex; height:3.009ex;" alt="{\displaystyle f\,'(0)=0}"></span>), lo nombre π se pòt definir coma lo pus pichon reau positiu <i>a</i> tau que cos(<i>a</i>)= -1. </p><p>Citem enfin (per clavar arbitrariament aquela lista) la definicion integrala seguenta de π, que se tròba dins cèrtei presentacions de l'analisi complèxa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =2\int _{-1}^{+1}{\frac {1}{t^{2}+1}}\ \mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =2\int _{-1}^{+1}{\frac {1}{t^{2}+1}}\ \mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6bca041149c47ae96d7c12537f792a987b47a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.61ex; height:6.343ex;" alt="{\displaystyle \pi =2\int _{-1}^{+1}{\frac {1}{t^{2}+1}}\ \mathrm {d} t}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Seguidas_e_serias">Seguidas e serias</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=10" title="Modificar la seccion : Seguidas e serias" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=10" title="Edita el codi de la secció: Seguidas e serias"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De seguidas o serias variadas convergisson vers π (o un sieu multiple racionau) e son la fònt de calculs d'aproximacions d'aqueu nombre. </p> <div class="mw-heading mw-heading4"><h4 id="Metòde_d’Arquimèdes"><span id="Met.C3.B2de_d.E2.80.99Arquim.C3.A8des"></span>Metòde d’Arquimèdes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=11" title="Modificar la seccion : Metòde d’Arquimèdes" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=11" title="Edita el codi de la secció: Metòde d’Arquimèdes"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{n\to +\infty }\left(n\cdot \sin \left({\frac {\pi }{n}}\right)\right)=\lim _{n\to +\infty }\left(n\cdot \tan \left({\frac {\pi }{n}}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{n\to +\infty }\left(n\cdot \sin \left({\frac {\pi }{n}}\right)\right)=\lim _{n\to +\infty }\left(n\cdot \tan \left({\frac {\pi }{n}}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd492b0eead56120bc6f3cf242310ae2480fcea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.332ex; height:5.009ex;" alt="{\displaystyle \pi =\lim _{n\to +\infty }\left(n\cdot \sin \left({\frac {\pi }{n}}\right)\right)=\lim _{n\to +\infty }\left(n\cdot \tan \left({\frac {\pi }{n}}\right)\right)}"></span>.</dd></dl> <p>Lei doas seguidas definidas per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle s_{n}=n\sin(\pi /n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s_{n}=n\sin(\pi /n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436237709adacf4b96906efa88514c96830cf2ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.436ex; height:2.176ex;" alt="{\displaystyle \scriptstyle s_{n}=n\sin(\pi /n)}"></span>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t_{n}=n\tan(\pi /n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t_{n}=n\tan(\pi /n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366d40f676d7b422e50967e0e327a14103f271da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.615ex; height:2.176ex;" alt="{\displaystyle \scriptstyle t_{n}=n\tan(\pi /n)}"></span>, <i>n</i> ≥ 3, representan lei semiperimètres dei <a href="/wiki/Polig%C3%B2n" title="Poligòn">poligòns</a> regulars de <i>n</i> costats, inscrich dins lo cercle trigonometric per <i>s</i><sub><i>n</i></sub>, exinscrich per <i>t</i><sub><i>n</i></sub>. S'utilizan per mejan de seguidas extrachas que son indèx (lo nombre de costats dau poligòn) dobla a cada iteracion, per obtenir π en passant au limit dins d’expressions ont apareisson leis operacions aritmeticas elementàrias e la <a href="/wiki/Rai%C3%A7_carrada" title="Raiç carrada">raiç carrada</a>. Ansin, òm se pòt inspirar dau metòde d'Arquimèdes — vejatz l'istoric dau calcul de π — per donar una definicion recurrenta dei seguidas extrachas qu'an per tèrmes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle s_{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s_{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f906e3a80ff52b60dfaa1e2eccaa1c49c3d20a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.537ex; height:1.676ex;" alt="{\displaystyle \scriptstyle s_{2^{n}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t_{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t_{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e4194d36169494e03f0175283f7c1da564d265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.359ex; height:2.009ex;" alt="{\displaystyle \scriptstyle t_{2^{n}}}"></span> o encara <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle s_{3.2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s_{3.2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a35a13fbcef39c1a845d423b638cf10a09322b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.575ex; height:1.676ex;" alt="{\displaystyle \scriptstyle s_{3.2^{n}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t_{3.2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t_{3.2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8caa925c26c81023a2969a9b0e39eddaa4f7049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.398ex; height:2.009ex;" alt="{\displaystyle \scriptstyle t_{3.2^{n}}}"></span>, gràcias ais <a href="/w/index.php?title=Identitat_trigonometrica&action=edit&redlink=1" class="new" title="Identitat trigonometrica (la pagina existís pas)">identitats trigonometricas</a> usualas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lll}t_{2n}=2{s_{n}\cdot t_{n} \over s_{n}+t_{n}}&t_{3}=3{\sqrt {3}}&t_{4}=4\\s_{2n}={\sqrt {s_{n}\cdot t_{2n}}}&s_{3}={3{\sqrt {3}} \over 2}&s_{4}={2{\sqrt {2}}}\,.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lll}t_{2n}=2{s_{n}\cdot t_{n} \over s_{n}+t_{n}}&t_{3}=3{\sqrt {3}}&t_{4}=4\\s_{2n}={\sqrt {s_{n}\cdot t_{2n}}}&s_{3}={3{\sqrt {3}} \over 2}&s_{4}={2{\sqrt {2}}}\,.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbf80c218e6f4506d05cd52faa9b4d3fefd80bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:40.221ex; height:8.676ex;" alt="{\displaystyle {\begin{array}{lll}t_{2n}=2{s_{n}\cdot t_{n} \over s_{n}+t_{n}}&t_{3}=3{\sqrt {3}}&t_{4}=4\\s_{2n}={\sqrt {s_{n}\cdot t_{2n}}}&s_{3}={3{\sqrt {3}} \over 2}&s_{4}={2{\sqrt {2}}}\,.\end{array}}}"></span></dd></dl> <p>En utilizant leis identitats trigonometricas, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 2\sin(x/2)={\sqrt {2-\cos(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2\sin(x/2)={\sqrt {2-\cos(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a03dcee47b85b700a1fd1d041b95aeefd743e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.534ex; height:2.676ex;" alt="{\displaystyle \scriptstyle 2\sin(x/2)={\sqrt {2-\cos(x)}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 2\cos(x/2)={\sqrt {2+\cos(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2\cos(x/2)={\sqrt {2+\cos(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de93bc75a1094a9447c9e8b5ef695d55454e3e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.714ex; height:2.676ex;" alt="{\displaystyle \scriptstyle 2\cos(x/2)={\sqrt {2+\cos(x)}}}"></span> (<i>x</i> ∈ [0,π]), se pòt exprimir <i>s</i><sub>2<sup><i>k+1</i></sup></sub> e <i>s</i><sub>3.2<sup><i>k</i></sup></sub> (k≥1) per embessonaments successius de raiç carradas. </p><p>Lo nombre π se pòt alora escriure coma una expression onte s'embessonan de raiç carradas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{k\to +\infty }\left(2^{k}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2}}}}}}}}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{k\to +\infty }\left(2^{k}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2}}}}}}}}}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098eda3bbc704bcd43645e5b77b9777feaa40b32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:56.338ex; height:10.843ex;" alt="{\displaystyle \pi =\lim _{k\to +\infty }\left(2^{k}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2}}}}}}}}}}}}\right)}"></span><br /></dd> <dd>(<i>k</i> es lo nombre de raiç carradas embessonadas)</dd></dl> <p>o encara: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{k\to +\infty }\left(3\cdot 2^{k-1}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{k\to +\infty }\left(3\cdot 2^{k-1}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6453004a0a3a1992b95e53e899718ddb6e6868f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:67.737ex; height:12.176ex;" alt="{\displaystyle \pi =\lim _{k\to +\infty }\left(3\cdot 2^{k-1}\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}}}}}\right)}"></span></dd></dl> <p>Una autra expression de <i>s</i><sub>2<sup><i>k+1</i></sup></sub>, que se pòt deduire simplament de la premiera d'aquelei doas egalitats (basta de multiplicar per √(2+√…)), mena au <a href="/w/index.php?title=Produch_infinit&action=edit&redlink=1" class="new" title="Produch infinit (la pagina existís pas)">produch infinit</a> seguent (formula de <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a>, <a href="/wiki/1593" class="mw-redirect" title="1593">1593</a>). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}={\frac {2}{\sqrt {2}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\cdot \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}={\frac {2}{\sqrt {2}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\cdot \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0199bb0f4534b89b453e5a1ef5ce61326bd743b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:43.81ex; height:9.343ex;" alt="{\displaystyle {\frac {\pi }{2}}={\frac {2}{\sqrt {2}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\cdot \cdots }"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Somas_e_produchs_infinits">Somas e produchs infinits</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=12" title="Modificar la seccion : Somas e produchs infinits" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=12" title="Edita el codi de la secció: Somas e produchs infinits"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{k}}{2k+1}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{k}}{2k+1}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}={\frac {\pi }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4943066ae40e6ef987d27059023c0ccdb7496b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.016ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{k}}{2k+1}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}={\frac {\pi }{4}}}"></span> (formula de <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>, <a href="/w/index.php?title=James_Gregory_(matematician)&action=edit&redlink=1" class="new" title="James Gregory (matematician) (la pagina existís pas)">James Gregory</a> e Madhava de Sangamagrama<sup id="cite_ref-LeibnizGregory_23-0" class="reference"><a href="#cite_note-LeibnizGregory-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-Madhava_24-0" class="reference"><a href="#cite_note-Madhava-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>)</li></ul> <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 {\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdot \cdots \cdot {\frac {2k+2}{2k+1}}\cdot {\frac {2k+2}{2k+3}}\cdot \cdots ={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>1</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</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>6</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>9</mn> </mfrac> </mrow> <mo>⋅</mo> <mo>⋯</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdot \cdots \cdot {\frac {2k+2}{2k+1}}\cdot {\frac {2k+2}{2k+3}}\cdot \cdots ={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236ccceaf97492fa54bfbd9abda323a5392558a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:59.597ex; height:5.676ex;" alt="{\displaystyle {\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdot \cdots \cdot {\frac {2k+2}{2k+1}}\cdot {\frac {2k+2}{2k+3}}\cdot \cdots ={\frac {\pi }{2}}}"></span> (produch de <a href="/wiki/John_Wallis#Produch_de_Wallis" title="John Wallis">Wallis</a>)</li></ul> <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 {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>9801</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mn>1103</mn> <mo>+</mo> <mn>26390</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mn>396</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776090fb01b361c6db0c8e97f61d8ca9911e435e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.871ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}"></span> (formula que se deu a <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan</a>)</li></ul> <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 {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mn>12</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>6</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mn>13591409</mn> <mo>+</mo> <mn>545140134</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mn>640320</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7303826e1e4fcba1e8f111880fa8b1b0c12c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.878ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}}"></span> (formula que se deu a <a href="/w/index.php?title=David_e_Gregory_Chudnovsky&action=edit&redlink=1" class="new" title="David e Gregory Chudnovsky (la pagina existís pas)">David e Gregory Chudnovsky</a>)</li></ul> <div class="mw-heading mw-heading4"><h4 id="Seguidas_recursivas">Seguidas recursivas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=13" title="Modificar la seccion : Seguidas recursivas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=13" title="Edita el codi de la secció: Seguidas recursivas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una seguida que sa definicion s'inspira de la <a href="/w/index.php?title=Formula_de_Brent-Salamin&action=edit&redlink=1" class="new" title="Formula de Brent-Salamin (la pagina existís pas)">formula de Brent-Salamin</a> (1975): </p><p>Sián tres seguidas realas <span class="mwe-math-element"><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})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9e36c3b62c43485a0a07aa4a4e71d9b4b1a8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.771ex; height:2.843ex;" alt="{\displaystyle (A_{n})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc23382987746e5ad84e68441c9eee4de8bf32a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.792ex; height:2.843ex;" alt="{\displaystyle (B_{n})}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (C_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e555618e681831ed3d8d7f41692b5c04df804c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.69ex; height:2.843ex;" alt="{\displaystyle (C_{n})}"></span> que se definisson mutualament: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ll}A_{0}=1&A_{n+1}={\frac {A_{n}+B_{n}}{2}}\\B_{0}={\sqrt {\frac {1}{2}}}&B_{n+1}={\sqrt {A_{n}\cdot B_{n}}}\\C_{0}={\frac {1}{4}}&C_{n+1}=C_{n}-2^{n}\left({\frac {A_{n}-B_{n}}{2}}\right)^{2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}A_{0}=1&A_{n+1}={\frac {A_{n}+B_{n}}{2}}\\B_{0}={\sqrt {\frac {1}{2}}}&B_{n+1}={\sqrt {A_{n}\cdot B_{n}}}\\C_{0}={\frac {1}{4}}&C_{n+1}=C_{n}-2^{n}\left({\frac {A_{n}-B_{n}}{2}}\right)^{2}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7c15dc8877aab4cb95bc11e9a00c1496f5a389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:39.508ex; height:14.843ex;" alt="{\displaystyle {\begin{array}{ll}A_{0}=1&A_{n+1}={\frac {A_{n}+B_{n}}{2}}\\B_{0}={\sqrt {\frac {1}{2}}}&B_{n+1}={\sqrt {A_{n}\cdot B_{n}}}\\C_{0}={\frac {1}{4}}&C_{n+1}=C_{n}-2^{n}\left({\frac {A_{n}-B_{n}}{2}}\right)^{2}\end{array}}}"></span></dd></dl> <p>Alara: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{n\to +\infty }{\frac {\left(A_{n}+B_{n}\right)^{2}}{4\cdot C_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <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> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>⋅</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{n\to +\infty }{\frac {\left(A_{n}+B_{n}\right)^{2}}{4\cdot C_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8300f8de61e0be504bbddf8a4b82f7a404955ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.853ex; height:6.509ex;" alt="{\displaystyle \pi =\lim _{n\to +\infty }{\frac {\left(A_{n}+B_{n}\right)^{2}}{4\cdot C_{n}}}}"></span></dd></dl> <p>Es de notar que lo nombre de decimalas exactas (en basa 10) dobla quasi a cada iteracion. </p> <div class="mw-heading mw-heading4"><h4 id="Foncion_zêta_de_Riemann"><span id="Foncion_z.C3.AAta_de_Riemann"></span>Foncion zêta de Riemann</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=14" title="Modificar la seccion : Foncion zêta de Riemann" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=14" title="Edita el codi de la secció: Foncion zêta de Riemann"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-section bandeau-niveau-detail loupe">Articles detalhats: <a href="/w/index.php?title=Probl%C3%A8ma_de_Basil%C3%A8a&action=edit&redlink=1" class="new" title="Problèma de Basilèa (la pagina existís pas)">Problèma de Basilèa</a> e <a href="/w/index.php?title=Foncion_z%C3%AAta_de_Riemann&action=edit&redlink=1" class="new" title="Foncion zêta de Riemann (la pagina existís pas)">Foncion zêta de Riemann</a>.</div> <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 \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{k^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{k^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21abbe22049e24157483328cda8e0ac0cdf1ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.29ex; height:6.176ex;" alt="{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{k^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}"></span> (<a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>)</li></ul> <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 \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots +{\frac {1}{k^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>90</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots +{\frac {1}{k^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4a790081ec74a2bab1e6aec2ad86ac31ab0ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.29ex; height:6.176ex;" alt="{\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots +{\frac {1}{k^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}}"></span>,</li></ul> <dl><dd>e pus generalament, Euler indica que <a href="/w/index.php?title=Foncion_z%C3%AAta_de_Riemann&action=edit&redlink=1" class="new" title="Foncion zêta de Riemann (la pagina existís pas)">ζ(2n)</a> es un multiple racionau de π<sup>2n</sup> per tot entier naturau <i>n</i>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Foncion_Gamma_d’Euler"><span id="Foncion_Gamma_d.E2.80.99Euler"></span>Foncion Gamma d’Euler</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=15" title="Modificar la seccion : Foncion Gamma d’Euler" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=15" title="Edita el codi de la secció: Foncion Gamma d’Euler"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \Gamma \left({\frac {1}{2}}\right)=\int _{0}^{+\infty }{\frac {\mathrm {e} ^{-t}}{\sqrt {t}}}\ \mathrm {d} t={\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> <msqrt> <mi>t</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left({\frac {1}{2}}\right)=\int _{0}^{+\infty }{\frac {\mathrm {e} ^{-t}}{\sqrt {t}}}\ \mathrm {d} t={\sqrt {\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a760c2a061fac3071500e9c3e8fa2e8fe43438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.415ex; height:6.509ex;" alt="{\displaystyle \Gamma \left({\frac {1}{2}}\right)=\int _{0}^{+\infty }{\frac {\mathrm {e} ^{-t}}{\sqrt {t}}}\ \mathrm {d} t={\sqrt {\pi }}}"></span> (<a href="/w/index.php?title=Foncion_Gamma_d%27Euler&action=edit&redlink=1" class="new" title="Foncion Gamma d'Euler (la pagina existís pas)">foncion gamma d'Euler</a>)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Probabilitats_e_estatisticas">Probabilitats e estatisticas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=16" title="Modificar la seccion : Probabilitats e estatisticas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=16" title="Edita el codi de la secció: Probabilitats e estatisticas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo nombre π apareis sovent en <a href="/w/index.php?title=Probabilitats&action=edit&redlink=1" class="new" title="Probabilitats (la pagina existís pas)">probabilitats</a> e en <a href="/w/index.php?title=Estatisticas&action=edit&redlink=1" class="new" title="Estatisticas (la pagina existís pas)">estatisticas</a>. Citem entre autrei: </p> <ul><li>l'integrala de Gauss, liada a la definicion de la <a href="/w/index.php?title=Lei_normala&action=edit&redlink=1" class="new" title="Lei normala (la pagina existís pas)">lei normala</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>:</li></ul> <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 \int _{-\infty }^{+\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7d2ce9c41e2c7aa676fef31ac1e8af17c35618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.36ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}"></span></dd></dl> <ul><li>La formula seguenta (<span class="citation">«<a href="/w/index.php?title=Formula_de_Stirling&action=edit&redlink=1" class="new" title="Formula de Stirling (la pagina existís pas)">formula de Stirling</a>»</span>), tirada de l’analisi, a d'aplicacions en probabilitats. En particular, permet de demostrar la convergéncia de la <a href="/w/index.php?title=Lei_binomiala&action=edit&redlink=1" class="new" title="Lei binomiala (la pagina existís pas)">lei binomiala</a> vèrs la <a href="/w/index.php?title=Lei_normala&action=edit&redlink=1" class="new" title="Lei normala (la pagina existís pas)">lei normala</a>:</li></ul> <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 n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977732acd9d1c6bbe89c54fc956f3c16d2feade9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.19ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"></span></dd> <dd>(significa que lo quocient dei dos membres a per limit 1).</dd></dl> <p>D'autra part, existís divèrseis experiéncias aleatòrias ont una probabilitat s'exprimís en foncion de π. Pòdon doncas servir (teoricament), en efectuant un grand nombre d'espròvas, a determinar una aproximacion de π. </p><p>L’<a href="/w/index.php?title=Agulha_de_Buffon&action=edit&redlink=1" class="new" title="Agulha de Buffon (la pagina existís pas)">agulha de Buffon</a> es una experiéncia aleatòria imaginada per lo naturalista <a href="/wiki/Georges-Louis_Leclerc,_comte_de_Buffon" title="Georges-Louis Leclerc, comte de Buffon">Buffon</a>. Consistís a mandar una agulha de longor <i>a</i> sus un postam que sei pòsts son de largor <i>L</i>. La question es de determinar la probabilitat que l’agulha cope la linha separant doas pòsts; aquela probabilitat es<sup id="cite_ref-bn_26-0" class="reference"><a href="#cite_note-bn-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>: </p> <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 p={\frac {2a}{\pi \times L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <mi>π</mi> <mo>×</mo> <mi>L</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {2a}{\pi \times L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf7a6d0c2b80f00627eb1ec99402e06d19350de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:10.949ex; height:5.176ex;" alt="{\displaystyle p={\frac {2a}{\pi \times L}}}"></span></dd></dl> <p>Se pòt utilizar aquò per determinar una aproximacion de π: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \approx {\frac {2na}{xL}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>a</mi> </mrow> <mrow> <mi>x</mi> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \approx {\frac {2na}{xL}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9ec65c7db763fbd2ec91a26b2c6894b7a6ac3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.7ex; height:5.176ex;" alt="{\displaystyle \pi \approx {\frac {2na}{xL}},}"></span></dd></dl> <p>onte <i>n</i> es lo nombre d'agulhas mandadas, e <i>x</i> es lo nombre d'aquelei que crosan una linha. D'efècte, se <i>n</i> es grand, la probabilitat <i>p</i> es vesina de la frequéncia <i>p' = x / n</i> (<a href="/w/index.php?title=Lei_dei_grands_nombres&action=edit&redlink=1" class="new" title="Lei dei grands nombres (la pagina existís pas)">lei dei grands nombres</a>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {2a}{Lp}}\approx {\frac {2a}{Lp'}}{\text{ e }}{\frac {2a}{Lp'}}={\frac {2na}{xL}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <mi>L</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <mi>L</mi> <msup> <mi>p</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> e </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <mi>L</mi> <msup> <mi>p</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>a</mi> </mrow> <mrow> <mi>x</mi> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {2a}{Lp}}\approx {\frac {2a}{Lp'}}{\text{ e }}{\frac {2a}{Lp'}}={\frac {2na}{xL}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e81a8c8dd9e918d0f92a5da1605c5a1a521947e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.226ex; height:5.676ex;" alt="{\displaystyle \pi ={\frac {2a}{Lp}}\approx {\frac {2a}{Lp'}}{\text{ e }}{\frac {2a}{Lp'}}={\frac {2na}{xL}}.}"></span></dd></dl> <p>La precision d'aqueu metòde es fòrça limitada; e mai lo resultat siá matematicament corrècte, se pòt pas utilizar per determinar experimentalament mai de quauquei decimalas de π<sup id="cite_ref-bn_26-1" class="reference"><a href="#cite_note-bn-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Monte-Carlo01.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Monte-Carlo01.gif/220px-Monte-Carlo01.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/ba/Monte-Carlo01.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Avaloracion de π per lo metòde de Montcarles.</figcaption></figure> <p>Una autra experiéncia aleatòria consistís a prendre <i>a l'azard</i> un ponch dins un carrat de costat 1; la probabilitat qu'aqueu ponch siá dins lo quart de disc de rai 1 vau π/4; aiçò es de bòn comprendre, que la superfícia dau quart de cercle es π/4 mentre que la superfícia dau carrat es 1. Se pòt simular aquela experiéncia aleatòria (<a href="/wiki/Met%C3%B2de_de_Montcarles" title="Metòde de Montcarles">metòde de Montcarles</a>) per avalorar π. </p> <div class="mw-heading mw-heading3"><h3 id="Seguida_logistica">Seguida logistica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=17" title="Modificar la seccion : Seguida logistica" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=17" title="Edita el codi de la secció: Seguida logistica"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Siá (<i>x</i><sub><i>n</i></sub>) la seguida deis iterats de la foncion logistica de paramètre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbc457cfb097c2933c8c525353d77382d21981a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =4}"></span> aplicada a un reau <i>x</i><sub><i>0</i></sub> chausit dins l'interval [0, 1] (valent a dire que se definís, per tot <span class="mwe-math-element"><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\geqslant 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geqslant 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0197a6a3f5aa0b8b9e4cc05f849b97c85c8f781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geqslant 0}"></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_{n+1}=4x_{n}(1-x_{n})~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=4x_{n}(1-x_{n})~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afab5fe51c86da716b578d327f5cb6d009b16a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.399ex; height:2.843ex;" alt="{\displaystyle x_{n+1}=4x_{n}(1-x_{n})~}"></span>).<br /> La seguida (<span class="mwe-math-element"><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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span>) sòrt de l'interval [0;1] e divergís per gaireben totei lei valors inicialas. </p><p>Òm 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 \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573193382da3d41b9475bcb7ebfde5432fbf7b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.674ex; height:6.843ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}\quad }"></span> per <a href="/w/index.php?title=Quasi_segurament&action=edit&redlink=1" class="new" title="Quasi segurament (la pagina existís pas)">quasi totei</a> lei valors inicialas <span class="mwe-math-element"><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"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietats_avançadas"><span id="Proprietats_avan.C3.A7adas"></span>Proprietats avançadas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=18" title="Modificar la seccion : Proprietats avançadas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=18" title="Edita el codi de la secció: Proprietats avançadas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Aproximacions_numericas">Aproximacions numericas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=19" title="Modificar la seccion : Aproximacions numericas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=19" title="Edita el codi de la secció: Aproximacions numericas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Coma π es transcendent, n'existís pas d'expression utilizant unicament de nombres e de foncions algebricas<sup id="cite_ref-ttop_11-1" class="reference"><a href="#cite_note-ttop-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>. Leis expressions de π ont apareisson ren que leis operacions de l'aritmetica elementària fan generalament intervenir de somas infinidas (serias) o de produchs infinits<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>; au mai s'aponde de tèrmes (o de factors) dins lo calcul, au mai lo resultat serà precís. </p><p>Per consequent, la preséncia de π dins de calculs numerics impausa de lo remplaçar per una aproximacion. Ben sovent, leis aproximacions 3,14 o 22/7 sufison; per una precision melhora, leis engenhaires utilizan sovent 3,1416 o 3,14159 (respectivament 5, 6 chifras significativas). Leis aproximacions 22/7 e 355/113 (qu'an respectivament 3, 6 chifras significativas), s'obtenon a partir dau desvolopament de π en <a href="/w/index.php?title=Fraccion_continua&action=edit&redlink=1" class="new" title="Fraccion continua (la pagina existís pas)">fraccion continua</a>. </p><p>L'aproximacion de π per 355/113 es la melhora que se pòsque escriure amb ren que 3 o 4 chifras au numerator e au denominator; la melhora aproximacion seguenta es 103993/33102, que ne demanda un nombre fòrça mai important (aiçò vèn de l'aparicion dau nombre 292 dins lo desvolopament en fraccion continua de π)<sup id="cite_ref-gourdon_31-0" class="reference"><a href="#cite_note-gourdon-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup>. </p><p>La premiera aproximacion numerica de π foguèt probable 3<sup id="cite_ref-ahop_32-0" class="reference"><a href="#cite_note-ahop-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>. Es una estimacion per defaut perqu'es lo repòrt entre lo perimètre d'un <a href="/wiki/Exag%C3%B2n" title="Exagòn">exagòn</a> regular inscrich dins un <a href="/wiki/Cercle" title="Cercle">cercle</a> e lo diamètre dau cercle. </p> <div class="mw-heading mw-heading3"><h3 id="Fraccions_continuas">Fraccions continuas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=20" title="Modificar la seccion : Fraccions continuas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=20" title="Edita el codi de la secció: Fraccions continuas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La seguida dei denominators parciaus dau desvolopament de π en <a href="/w/index.php?title=Fraccion_continua&action=edit&redlink=1" class="new" title="Fraccion continua (la pagina existís pas)">fraccion continua</a> a pas de regularitat vesedoira<sup id="cite_ref-ReferenceA_33-0" class="reference"><a href="#cite_note-ReferenceA-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>7</mn> <mo>,</mo> <mn>15</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>292</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>14</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>84</mn> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec4d08b565377b9b786c48d09bd265d62339d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.58ex; height:2.843ex;" alt="{\displaystyle \pi =[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots ]}"></span>,</dd></dl> <p>çò qu'es una notacion equivalenta a: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{3+\textstyle {\frac {1}{1+\textstyle {\frac {1}{14+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{84+\cdots }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>7</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>292</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>14</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>84</mn> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{3+\textstyle {\frac {1}{1+\textstyle {\frac {1}{14+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{84+\cdots }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699aef218f939f5851ff85e9e7612e259c56777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -41.338ex; width:76.139ex; height:43.676ex;" alt="{\displaystyle \pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{3+\textstyle {\frac {1}{1+\textstyle {\frac {1}{14+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{2+\textstyle {\frac {1}{1+\textstyle {\frac {1}{84+\cdots }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}"></span></dd></dl> <p>Pasmens, existís de fraccions continuas generalizadas representant π e qu'an una estructura regulara<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\textstyle {\frac {11^{2}}{2+\cdots }}}}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\textstyle {\frac {11^{2}}{6+\cdots }}}}}}}}}}}}=\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\textstyle {\frac {5^{2}}{11+\cdots }}}}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>9</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>11</mn> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\textstyle {\frac {11^{2}}{2+\cdots }}}}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\textstyle {\frac {11^{2}}{6+\cdots }}}}}}}}}}}}=\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\textstyle {\frac {5^{2}}{11+\cdots }}}}}}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a78aa1b85edab062e36fd55ef6a67a9cb4086b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.838ex; width:77.026ex; height:18.676ex;" alt="{\displaystyle \pi =\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\textstyle {\frac {11^{2}}{2+\cdots }}}}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\textstyle {\frac {11^{2}}{6+\cdots }}}}}}}}}}}}=\textstyle {\frac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\textstyle {\frac {5^{2}}{11+\cdots }}}}}}}}}}}}}"></span></dd></dl> <p>Lo nombre π/2 se pòt tanben escriure coma fraccion continua generalizada, fasent intervenir la seguida deis invèrs dei nombres entiers: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}=1+{\frac {1}{1+{\frac {1}{1/2+{\frac {1}{1/3+\,\cdots +{\frac {1}{1/n+\,\cdots }}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}=1+{\frac {1}{1+{\frac {1}{1/2+{\frac {1}{1/3+\,\cdots +{\frac {1}{1/n+\,\cdots }}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0101451d95af6d4286fa785a2ef96e1cc66c3942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:31.514ex; height:10.509ex;" alt="{\displaystyle {\frac {\pi }{2}}=1+{\frac {1}{1+{\frac {1}{1/2+{\frac {1}{1/3+\,\cdots +{\frac {1}{1/n+\,\cdots }}}}}}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Questions_dubèrtas"><span id="Questions_dub.C3.A8rtas"></span>Questions dubèrtas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=21" title="Modificar la seccion : Questions dubèrtas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=21" title="Edita el codi de la secció: Questions dubèrtas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De questions nombrosas se pausan encara: π, e son dos nombres transcendents, mai son algebricament independents o existís un polinòmi de doas variablas e de coeficients entiers que lo pareu (π, e) ne siá solucion? La question es encara en suspens. En 1929, <a href="/w/index.php?title=Aleksandr_Gelfond&action=edit&redlink=1" class="new" title="Aleksandr Gelfond (la pagina existís pas)">Aleksandr Gelfond</a> demostrèt que e<sup>π</sup> es transcendent<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> e en 1996, Iurii Nesterenko a demostrat que π e e<sup>π</sup> son algebricament independents. </p><p>Coma es estat dich <a href="#Représentation_decimala"><i>supra</i></a>, se sap pas encara se lo nombre π es <a href="/w/index.php?title=Nombre_normau&action=edit&redlink=1" class="new" title="Nombre normau (la pagina existís pas)">normau</a>, ni mai s'es un <a href="/w/index.php?title=Nombre_univ%C3%A8rs&action=edit&redlink=1" class="new" title="Nombre univèrs (la pagina existís pas)">nombre univèrs</a> en <a href="/w/index.php?title=Sist%C3%A8ma_decimau&action=edit&redlink=1" class="new" title="Sistèma decimau (la pagina existís pas)">basa 10</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Lo_nombre_π_dins_l'art"><span id="Lo_nombre_.CF.80_dins_l.27art"></span>Lo nombre π dins l'art</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=22" title="Modificar la seccion : Lo nombre π dins l'art" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=22" title="Edita el codi de la secció: Lo nombre π dins l'art"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fòrça obratges o sites senhalan la preséncia supausada dau nombre π dins l'arquitectura dei piramidas; pus precisament, π seriá lo repòrt entre lo perimètre de la basa e lo doble de l'autor dei piramidas<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup>. Es verai que la penda de la <a href="/wiki/Piramida_de_Kheops" class="mw-redirect" title="Piramida de Kheops">piramida de Kheops</a> es de 14/11; per tant, lo repòrt de la basa a l'autor es 22/14, la mitat de 22/7, una aproximacion frequenta de π. Per aquò, i fau veire una intencion? Es mai que dobtós<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> puei que la penda dei piramidas es pas constanta e que, segon lei regions e leis epòcas, se tròba de pendas de 6/5 (<a href="/w/index.php?title=Piramida_roja&action=edit&redlink=1" class="new" title="Piramida roja (la pagina existís pas)">piramida roja</a>), 4/3 (<a href="/w/index.php?title=Piramida_de_Khephren&action=edit&redlink=1" class="new" title="Piramida de Khephren (la pagina existís pas)">piramida de Khephren</a>) o 7/5 (<a href="/wiki/Piramida_rombo%C3%AFdala" title="Piramida romboïdala">piramida romboïdala</a>) que menan a un repòrt (entre perimètre e doble de l'autor) alunhat de π. </p><p>En tot cas, lo nombre π es present dins la cultura artistica modèrna. Per exemple, dins <i>Contact</i>, un roman de <a href="/wiki/Carl_Sagan" title="Carl Sagan">Carl Sagan</a>, jòga un ròtle clau dins lo scenario e se suggerís que i a un messatge escondut fonsament dins sei decimalas, plaçat per lo creator de l'univèrs. Aquela partida de l'istòria es estada levada de l'adaptacion dau roman au cinèma. </p><p>Dins lo domeni musicau, la cantairitz e compositritz <a href="/wiki/Kate_Bush" title="Kate Bush">Kate Bush</a> publiquèt en 2005 son album <i>Aerial</i>, contenent lo tròç « π » que sei paraulas son principalament compausadas dei decimalas de π<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Nòtas_e_referéncias"><span id="N.C3.B2tas_e_refer.C3.A9ncias"></span>Nòtas e referéncias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=23" title="Modificar la seccion : Nòtas e referéncias" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=23" title="Edita el codi de la secció: Nòtas e referéncias"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><sup id="cite_ref-ReferenceA_33-1" class="reference"><a href="#cite_note-ReferenceA-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ahop_32-1" class="reference"><a href="#cite_note-ahop-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="references-small decimal" style="-moz-column-count:2;-webkit-column-count:2;column-count:2;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> An Introduction to the History of Mathematics (en anglés).  Holt, Rinehart & Winston.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-collier-2"><span class="mw-cite-backlink"><a href="#cite_ref-collier_2-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> Pi (Collier's Encyclopedia) (en anglés). 19.  New York: Macmillan Educational Corporation.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-adm-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-adm_3-0">3,0</a></sup> et <sup><a href="#cite_ref-adm_3-1">3,1</a></sup></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <style data-mw-deduplicate="TemplateStyles:r2450626">.mw-parser-output span.default a{background:url("//upload.wikimedia.org/wikipedia/commons/0/00/Lock_icon_blue.gif")center right no-repeat!important;padding-right:17px!important}.mw-parser-output span.abonament a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")center right no-repeat!important;padding-right:17px!important}.mw-parser-output span.liure a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")center right no-repeat!important;padding-right:17px!important}.mw-parser-output span.limitat a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg")center right no-repeat!important;padding-right:17px!important}.mw-parser-output span.pdf a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Icons-mini-file_acrobat.gif/15px-Icons-mini-file_acrobat.gif")center right no-repeat!important;padding-right:17px!important}</style><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://mathforum.org/dr.math/faq/faq.pi.html">About Pi</a></span>».  Ask Dr. Math FAQ.</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Bettina Richmond. «<span class="default"><a rel="nofollow" class="external text" href="http://www.wku.edu/~tom.richmond/Pir2.html">Area of a Circle</a></span>».  Western Kentucky University.</span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> Principles of Mathematical Analysis (en anglés).  McGraw-Hill. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/0-07-054235-X" title="Especial:Obratge de referéncia/0-07-054235-X">ISBN 0-07-054235-X</a></span>.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span><a rel="nofollow" class="external text" href="http://www.muslimheritage.com/topics/default.cfm?TaxonomyTypeID=12&TaxonomySubTypeID=59&TaxonomyThirdLevelID=-1&ArticleID=997">Glimpses in the history of a great number: Pi in Arabic mathematics</a> per Mustafa Mawaldi</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> Mémoire sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques (en anglés). XVII.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><span class="ouvrage" id="Niven1947"><span class="ouvrage" id="Ivan_Niven1947"><span class="indicador-lenga">(<abbr class="abbr" title="Lenga : anglés">en</abbr>)</span> Ivan <span class="nom_auteur">Niven</span>, <span lang="en">« <cite style="font-style:normal;">A simple proof that π is irrational</cite> »</span>, <i><span class="lang-en" lang="en">Bulletin of the American Mathematical Society</span></i>, <abbr class="abbr" title="volum">vol.</abbr> 53, <abbr class="abbr" title="numèro">n<sup>o</sup></abbr> 6,‎ <time>1947</time>, <abbr class="abbr" title="pagina(s)">p.</abbr> 509 <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.ams.org/bull/1947-53-06/S0002-9904-1947-08821-2/S0002-9904-1947-08821-2.pdf">legir en linha</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+simple+proof+that+%CF%80+is+irrational&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.eissida=6&rft.aulast=Niven&rft.aufirst=Ivan&rft.volume=53&rfr_id=info%3Asid%2Foc.wikipedia.org%3APi" id="COinS_27375"></span></span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Helmut Richter. «<span class="default"><a rel="nofollow" class="external text" href="http://www.lrz-muenchen.de/~hr/numb/pi-irr.html">Pi Is Irrational</a></span>».  Leibniz Rechenzentrum.</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> Scientific Inference (en anglés).  Cambridge University Press.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-ttop-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ttop_11-0">11,0</a></sup> et <sup><a href="#cite_ref-ttop_11-1">11,1</a></sup></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Steve Mayer. «<span class="default"><a rel="nofollow" class="external text" href="http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html">The Transcendence of π</a></span>».</span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/impossible/sq_circle.shtml">Squaring the Circle</a></span>».  cut-the-knot.</span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://www.super-computing.org/pi_current.html">Current publicized world record of pi</a></span>».</span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> Excursions in Calculus (en anglés).  Washington: Mathematical Association of America (MAA). <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/0883853175" title="Especial:Obratge de referéncia/0883853175">ISBN 0883853175</a></span>.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000004000298000001&idtype=cvips&gifs=yes">Statistical estimacion of pi using random vectors</a></span>».</span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PiDigits.html">Pi Digits</a></span>». <i>MathWorld</i>.</span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text"><span class="ouvrage" id="Boutin2005"><span class="ouvrage" id="Chad_Boutin2005"><span class="indicador-lenga">(<abbr class="abbr" title="Lenga : anglés">en</abbr>)</span> Chad <span class="nom_auteur">Boutin</span>, <span lang="en">« <cite style="font-style:normal;">Pi seems a good random number generator - but not always the best</cite> »</span>, <i><span class="lang-en" lang="en">Purdue University</span></i>,‎ <time>2005</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.purdue.edu/UNS/html4ever/2005/050426.Fischbach.pi.html">legir en linha</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Pi+seems+a+good+random+number+generator+-+but+not+always+the+best&rft.jtitle=Purdue+University&rft.aulast=Boutin&rft.aufirst=Chad&rfr_id=info%3Asid%2Foc.wikipedia.org%3APi" id="COinS_24938"></span></span></span></span> </li> <li id="cite_note-Conférence_Delahaye-18"><span class="mw-cite-backlink"><a href="#cite_ref-Conférence_Delahaye_18-0">↑</a></span> <span class="reference-text"> Conférence de Jean-Paul Delahaye, <i>le nombre pi est-il simple o compliqué ?</i>, mardi 3 octobre 2006, cité des sciences, consultable <a rel="nofollow" class="external text" href="http://www.universcience.fr/fr/conferences-du-college/seance/c/1239026849604/le-nombre-pi-est-il-simple-ou-complique--/p/1239022827777/">aicí</a> </span> </li> <li id="cite_note-NOVA-19"><span class="mw-cite-backlink"><a href="#cite_ref-NOVA_19-0">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Rick Groleau. «<span class="default"><a rel="nofollow" class="external text" href="http://www.pbs.org/wgbh/nova/archimedes/pi.html">Infinite Secrets: Approximating Pi</a></span>».  NOVA.</span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> A History of Pi (en anglés).  Barnes & Noble Publishing. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/0880294183" title="Especial:Obratge de referéncia/0880294183">ISBN 0880294183</a></span>.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> The Number π (en anglés).  American Mathematical Society. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/0821832468" title="Especial:Obratge de referéncia/0821832468">ISBN 0821832468</a></span>.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text"><span class="ouvrage" id="Lampret2006"><span class="ouvrage" id="Vito_Lampret2006"><span class="indicador-lenga">(<abbr class="abbr" title="Lenga : anglés">en</abbr>)</span> Vito <span class="nom_auteur">Lampret</span>, <span lang="en">« <cite style="font-style:normal;">Even from Gregory-Leibniz series π could be computed: an example of how convergence of series can be accelerated</cite> »</span>, <i><span class="lang-en" lang="en">Lecturas Mathematicas</span></i>,‎ <time>2006</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.scm.org.co/Articulos/832.pdf">legir en linha</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Even+from+Gregory-Leibniz+series+%CF%80+could+be+computed%3A+an+example+of+how+convergence+of+series+can+be+accelerated&rft.jtitle=Lecturas+Mathematicas&rft.aulast=Lampret&rft.aufirst=Vito&rfr_id=info%3Asid%2Foc.wikipedia.org%3APi" id="COinS_30246"></span></span></span></span> </li> <li id="cite_note-LeibnizGregory-23"><span class="mw-cite-backlink"><a href="#cite_ref-LeibnizGregory_23-0">↑</a></span> <span class="reference-text">atribuida sovent a <a href="/w/index.php?title=Leibniz&action=edit&redlink=1" class="new" title="Leibniz (la pagina existís pas)">Leibniz</a>, mai probable que foguèt descubèrta anteriorament per Gregory, cf. <span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_through_the_ages.html"><span class="lang-en" lang="en"><i>Pi_through_the_ages.html</i></span></a> sus lo site de l’Universitat de Saint Andrews. Aquela formula èra pereu estada trobada devèrs 1400 per lo matematician indian <a href="/w/index.php?title=Madhava_de_Sangamagrama&action=edit&redlink=1" class="new" title="Madhava de Sangamagrama (la pagina existís pas)">Madhava</a>, mai aquela descubèrta demorèt inconeguda dau monde occidentau.</span> </li> <li id="cite_note-Madhava-24"><span class="mw-cite-backlink"><a href="#cite_ref-Madhava_24-0">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Madhava.html">Biographie de Madhava</a> sus lo site de l’Universitat de Saint-Andrew</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><a href="#cite_ref-25">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Eric W. Weisstein. «<span class="default"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/GaussianIntegral.html">Gaussian Integral</a></span>».  MathWorld.</span></span> </li> <li id="cite_note-bn-26"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-bn_26-0">26,0</a></sup> et <sup><a href="#cite_ref-bn_26-1">26,1</a></sup></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Eric W. Weisstein. «<span class="default"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BuffonsNeedleProblem.html">Buffon's Needle Problem</a></span>».  MathWorld.</span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><a href="#cite_ref-27">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Alex Bogomolny. «<span class="default"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/ctk/August2001.shtml">Math Surprises: An Example</a></span>».  cut-the-knot.</span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><a href="#cite_ref-28">↑</a></span> <span class="reference-text"><span class="ouvrage" id="Ramaley1969"><span class="ouvrage" id="J._F._Ramaley1969"><span class="indicador-lenga">(<abbr class="abbr" title="Lenga : anglés">en</abbr>)</span> J. F. <span class="nom_auteur">Ramaley</span>, <span lang="en">« <cite style="font-style:normal;">Buffon's Noodle Problem</cite> »</span>, <i><span class="lang-en" lang="en">The American Mathematical Monthly</span></i>,‎ <time>1969</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Buffon%27s+Noodle+Problem&rft.jtitle=The+American+Mathematical+Monthly&rft.aulast=Ramaley&rft.aufirst=J.+F.&rfr_id=info%3Asid%2Foc.wikipedia.org%3APi" id="COinS_22507"></span></span></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><a href="#cite_ref-29">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">«<span class="default"><a rel="nofollow" class="external text" href="http://www.datastructures.info/the-monte-carlo-algorithmmethod/">The Monte Carlo algorithm/method</a></span>». <i>datastructures</i>.</span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><a href="#cite_ref-30">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Eric W. Weisstein. «<span class="default"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PiFormulas.html">Pi Formulas</a></span>».  MathWorld.</span></span> </li> <li id="cite_note-gourdon-31"><span class="mw-cite-backlink"><a href="#cite_ref-gourdon_31-0">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">Xavier Gourdon. «<span class="default"><a rel="nofollow" class="external text" href="http://numbers.computation.free.fr/Constants/Pi/piApprox.html">Collection of aproximacions for π</a></span>».  Numbers, constants and computation.</span></span> </li> <li id="cite_note-ahop-32"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ahop_32-0">32,0</a></sup> et <sup><a href="#cite_ref-ahop_32-1">32,1</a></sup></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">.</span></span> </li> <li id="cite_note-ReferenceA-33"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ReferenceA_33-0">33,0</a></sup> et <sup><a href="#cite_ref-ReferenceA_33-1">33,1</a></sup></span> <span class="reference-text">[[OEIS:{{{id}}}|{{{id}}}]]: Continued fraccion for Pi, <a href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&redlink=1" class="new" title="On-Line Encyclopedia of Integer Sequences (la pagina existís pas)">On-Line Encyclopedia of Integer Sequences</a></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><a href="#cite_ref-34">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREF"> An Elegant Continued Fraction for <i>π</i> (en anglés). 106.  The American Mathematical Monthly.</span><span style="display: none;"> </span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><a href="#cite_ref-35">↑</a></span> <span class="reference-text">La Recherche, <span style="cursor:help;" title="numèro">n<sup>o</sup> </span>392, Décembre 2005,<i> L'indispensable nombre π</i></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><a href="#cite_ref-36">↑</a></span> <span class="reference-text"> Vejatz per exemple <i>Le secret de la grande pyramide</i> de George Barbarin</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><a href="#cite_ref-37">↑</a></span> <span class="reference-text"> Segon <i>The journal of the Society for the study of Egyptian Antiquities</i>, ISSN 0383-9753, 1978, vol 8, n4, « la valor de π qu'apareis dins la relacion entre l'autor e la longor de la piramida es probable fortuita »</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><a href="#cite_ref-38">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2450626"><span class="citation noarchive" style="font-style:normal" id="CITEREF">David Blatner. «<span class="default"><a rel="nofollow" class="external text" href="http://news.bbc.co.uk/1/hi/magazine/7296224.stm">UK | Magazine | 3.14 and the rest</a></span>».  BBC News.</span></span> </li> </ol> </div> <div class="mw-heading mw-heading3"><h3 id="Liames_extèrnes"><span id="Liames_ext.C3.A8rnes"></span>Liames extèrnes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pi&veaction=edit&section=24" title="Modificar la seccion : Liames extèrnes" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pi&action=edit&section=24" title="Edita el codi de la secció: Liames extèrnes"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En francés">(<abbr title="">fr</abbr>)</span> La preuve par Lambert de l’irrationnalité de π (1761), commentée sur lo site <a rel="nofollow" class="external text" href="http://www.bibnum.education.fr/mathematiques/lambert-et-l%E2%80%99irrationalite-de-p-1761">BibNum</a></li> <li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En francés">(<abbr title="">fr</abbr>)</span> De nombreuses informations historiques et mathématiques sur pi dans <a rel="nofollow" class="external text" href="http://www.pi314.net/">pi314.net</a></li> <li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <a rel="nofollow" class="external text" href="http://www.angio.net/pi/piquery">Site permetent una recèrca de chifras dins lei 200 000 000 premierei decimalas</a></li> <li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PiFormulas.html">Lo site Wolfram Mathematics</a> acampa un grand nombre de formulas pertocant π</li> <li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100317134116/http://www.math.utah.edu/~pa/math/pi.html">10 000 decimalas de π</a></li></ul> <table class="navbox collapsible noprint autocollapse" data-autocollapse-group="paleta" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinksneverexpand nowrap tnavbar" style="background-color:transparent; padding:0; font-size:xx-small; color:#000000;">[[Modèl:{{{modèl}}}|<abbr class="abbr" title="Veire aqueste modèl.">v</abbr>]] · [{{fullurl:Modèl:{{{modèl}}}|action=edit}} <abbr class="abbr" title="Modificar aqueste modèl. Mercé de previsualizar abans de salvar.">m</abbr>]</div></div><span style="font-size:110%">{{{títol}}}</span></th> </tr> </tbody></table>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Recuperada de « <a dir="ltr" href="https://oc.wikipedia.org/w/index.php?title=Pi&oldid=2438332">https://oc.wikipedia.org/w/index.php?title=Pi&oldid=2438332</a> »</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias de la pagina</a> : <ul><li><a href="/w/index.php?title=Categoria:P%C3%A0gines_amb_mides_d%27imatge_que_contenen_px_addicionals&action=edit&redlink=1" class="new" title="Categoria:Pàgines amb mides d'imatge que contenen px addicionals (la pagina existís pas)">Pàgines amb mides d'imatge que contenen px addicionals</a></li><li><a href="/wiki/Categoria:Mod%C3%A8l_infobox" title="Categoria:Modèl infobox">Modèl infobox</a></li><li><a href="/wiki/Categoria:Nombre_transcendent" title="Categoria:Nombre transcendent">Nombre transcendent</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria amagada : <ul><li><a href="/wiki/Categoria:Pagina_utilizant_lo_mod%C3%A8l_Refer%C3%A9ncias_amb_un_param%C3%A8tre_obsol%C3%A8t" title="Categoria:Pagina utilizant lo modèl Referéncias amb un paramètre obsolèt">Pagina utilizant lo modèl Referéncias amb un paramètre obsolèt</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"> Aquela pagina es estada modificada pel darrièr còp lo 15 mai de 2024 a 09.02.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A8dia:Drech_d%27autor" title="Wikipèdia:Drech d'autor">Drech d'autor</a> : Los tèxtes son disponibles jos <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.fr">licéncia Creative Commons paternitat pertatge a l’identic</a> ; d’autras condicions se pòdon aplicar. 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