Pii (vakio) – Wikipedia

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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">piilota</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">Johdanto</div> </a> </li> <li id="toc-Piin_vanhoja_likiarvoja" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piin_vanhoja_likiarvoja"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Piin vanhoja likiarvoja</span> </div> </a> <ul id="toc-Piin_vanhoja_likiarvoja-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piin_laskeminen_sarjakehitelmien_avulla" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piin_laskeminen_sarjakehitelmien_avulla"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Piin laskeminen sarjakehitelmien avulla</span> </div> </a> <ul id="toc-Piin_laskeminen_sarjakehitelmien_avulla-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piin_laskeminen_tulokehitelmien_avulla" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piin_laskeminen_tulokehitelmien_avulla"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Piin laskeminen tulokehitelmien avulla</span> </div> </a> <ul id="toc-Piin_laskeminen_tulokehitelmien_avulla-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piin_approksimaatioita" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piin_approksimaatioita"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Piin approksimaatioita</span> </div> </a> <ul id="toc-Piin_approksimaatioita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piin_desimaalien_laskeminen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Piin_desimaalien_laskeminen"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Piin desimaalien laskeminen</span> </div> </a> <ul id="toc-Piin_desimaalien_laskeminen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Muita_esimerkkejä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Muita_esimerkkejä"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Muita esimerkkejä</span> </div> </a> <ul id="toc-Muita_esimerkkejä-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Avoimia_kysymyksiä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Avoimia_kysymyksiä"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Avoimia kysymyksiä</span> </div> </a> <ul id="toc-Avoimia_kysymyksiä-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Katso_myös" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Katso_myös"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Katso myös</span> </div> </a> <ul id="toc-Katso_myös-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lähteet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lähteet"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Lähteet</span> </div> </a> <button aria-controls="toc-Lähteet-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>Vaihda alaosio Lähteet</span> </button> <ul id="toc-Lähteet-sublist" class="vector-toc-list"> <li id="toc-Viitteet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Viitteet"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Viitteet</span> </div> </a> <ul id="toc-Viitteet-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kirjallisuutta" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kirjallisuutta"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Kirjallisuutta</span> </div> </a> <ul id="toc-Kirjallisuutta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aiheesta_muualla" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aiheesta_muualla"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Aiheesta muualla</span> </div> </a> <ul id="toc-Aiheesta_muualla-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sisällysluettelo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sisällysluettelo" > <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="Vaihda sisällysluettelo" > <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">Vaihda sisällysluettelo</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">Pii (vakio)</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="Mene artikkeliin toisella kielellä. Saatavilla 162 kielellä" > <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 kieltä</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="suositeltu artikkeli"><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) — sveitsinsaksa" lang="gsw" hreflang="gsw" data-title="Pi (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="sveitsinsaksa" 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 — amhara" lang="am" hreflang="am" data-title="ፓይ" data-language-autonym="አማርኛ" data-language-local-name="amhara" class="interlanguage-link-target"><span>አማርኛ</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) — arabia" lang="ar" hreflang="ar" data-title="ط (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabia" class="interlanguage-link-target"><span>العربية</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 — aragonia" lang="an" hreflang="an" data-title="Numero π" data-language-autonym="Aragonés" data-language-local-name="aragonia" class="interlanguage-link-target"><span>Aragonés</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 — assami" lang="as" hreflang="as" data-title="পাই" data-language-autonym="অসমীয়া" data-language-local-name="assami" 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 — asturia" lang="ast" hreflang="ast" data-title="Númberu π" data-language-autonym="Asturianu" data-language-local-name="asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Pi" title="Pi — guarani" lang="gn" hreflang="gn" data-title="Pi" data-language-autonym="Avañe'ẽ" data-language-local-name="guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Pi" title="Pi — azeri" lang="az" hreflang="az" data-title="Pi" data-language-autonym="Azərbaycanca" data-language-local-name="azeri" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pi" title="Pi — indonesia" lang="id" hreflang="id" data-title="Pi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pi" title="Pi — malaiji" lang="ms" hreflang="ms" data-title="Pi" data-language-autonym="Bahasa Melayu" data-language-local-name="malaiji" class="interlanguage-link-target"><span>Bahasa Melayu</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 — bengali" lang="bn" hreflang="bn" data-title="পাই" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</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-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 — min nan -kiina" lang="nan" hreflang="nan" data-title="Îⁿ-chiu-lu̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan -kiina" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9F%D0%B8_(%D2%BB%D0%B0%D0%BD)" title="Пи (һан) — baškiiri" lang="ba" hreflang="ba" data-title="Пи (һан)" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%96" title="Пі — valkovenäjä" lang="be" hreflang="be" data-title="Пі" data-language-autonym="Беларуская" data-language-local-name="valkovenäjä" 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-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-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Pi" title="Pi — bosnia" lang="bs" hreflang="bs" data-title="Pi" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Pi_(niver)" title="Pi (niver) — bretoni" lang="br" hreflang="br" data-title="Pi (niver)" data-language-autonym="Brezhoneg" data-language-local-name="bretoni" class="interlanguage-link-target"><span>Brezhoneg</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 — bulgaria" lang="bg" hreflang="bg" data-title="Пи" data-language-autonym="Български" data-language-local-name="bulgaria" class="interlanguage-link-target"><span>Български</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="suositeltu artikkeli"><a href="https://ca.wikipedia.org/wiki/Nombre_%CF%80" title="Nombre π — katalaani" lang="ca" hreflang="ca" data-title="Nombre π" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Pi" title="Pi — cebuano" lang="ceb" hreflang="ceb" data-title="Pi" data-language-autonym="Cebuano" data-language-local-name="cebuano" class="interlanguage-link-target"><span>Cebuano</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) — tšuvassi" lang="cv" hreflang="cv" data-title="Пи (хисеп)" data-language-autonym="Чӑвашла" data-language-local-name="tšuvassi" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://cs.wikipedia.org/wiki/P%C3%AD_(%C4%8D%C3%ADslo)" title="Pí (číslo) — tšekki" lang="cs" hreflang="cs" data-title="Pí (číslo)" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pi_(mathemateg)" title="Pi (mathemateg) — kymri" lang="cy" hreflang="cy" data-title="Pi (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="kymri" 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) — tanska" lang="da" hreflang="da" data-title="Pi (tal)" data-language-autonym="Dansk" data-language-local-name="tanska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://de.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl — saksa" lang="de" hreflang="de" data-title="Kreiszahl" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</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 — alasorbi" lang="dsb" hreflang="dsb" data-title="Konstanta π" data-language-autonym="Dolnoserbski" data-language-local-name="alasorbi" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pii" title="Pii — viro" lang="et" hreflang="et" data-title="Pii" data-language-autonym="Eesti" data-language-local-name="viro" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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) — kreikka" lang="el" hreflang="el" data-title="Π (μαθηματική σταθερά)" data-language-autonym="Ελληνικά" data-language-local-name="kreikka" 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="suositeltu artikkeli"><a href="https://en.wikipedia.org/wiki/Pi" title="Pi — englanti" lang="en" hreflang="en" data-title="Pi" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</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 — espanja" lang="es" hreflang="es" data-title="Número π" data-language-autonym="Español" data-language-local-name="espanja" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><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-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/N%C3%BAmiru_%CF%80" title="Númiru π — extremadura" lang="ext" hreflang="ext" data-title="Númiru π" data-language-autonym="Estremeñu" data-language-local-name="extremadura" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Pi_(zenbakia)" title="Pi (zenbakia) — baski" lang="eu" hreflang="eu" data-title="Pi (zenbakia)" data-language-autonym="Euskara" data-language-local-name="baski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%BE%DB%8C" title="عدد پی — persia" lang="fa" hreflang="fa" data-title="عدد پی" data-language-autonym="فارسی" data-language-local-name="persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Pi" title="Pi — fidžinhindi" lang="hif" hreflang="hif" data-title="Pi" data-language-autonym="Fiji Hindi" data-language-local-name="fidžinhindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Pi" title="Pi — fääri" lang="fo" hreflang="fo" data-title="Pi" data-language-autonym="Føroyskt" data-language-local-name="fääri" 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 — ranska" lang="fr" hreflang="fr" data-title="Pi" data-language-autonym="Français" data-language-local-name="ranska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Py_(wiskunde)" title="Py (wiskunde) — länsifriisi" lang="fy" hreflang="fy" data-title="Py (wiskunde)" data-language-autonym="Frysk" data-language-local-name="länsifriisi" class="interlanguage-link-target"><span>Frysk</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 — friuli" lang="fur" hreflang="fur" data-title="Pi grêc" data-language-autonym="Furlan" data-language-local-name="friuli" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/P%C3%AD_(uimhir)" title="Pí (uimhir) — iiri" lang="ga" hreflang="ga" data-title="Pí (uimhir)" data-language-autonym="Gaeilge" data-language-local-name="iiri" class="interlanguage-link-target"><span>Gaeilge</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) — gaeli" lang="gd" hreflang="gd" data-title="Pi (àireamh)" data-language-autonym="Gàidhlig" data-language-local-name="gaeli" 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 — galicia" lang="gl" hreflang="gl" data-title="Número pi" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 — gan-kiina" lang="gan" hreflang="gan" data-title="圓周率" data-language-autonym="贛語" data-language-local-name="gan-kiina" class="interlanguage-link-target"><span>贛語</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 — gudžarati" lang="gu" hreflang="gu" data-title="પાઇ" data-language-autonym="ગુજરાતી" data-language-local-name="gudžarati" class="interlanguage-link-target"><span>ગુજરાતી</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 — kalmukki" lang="xal" hreflang="xal" data-title="Пи" data-language-autonym="Хальмг" data-language-local-name="kalmukki" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율 — korea" lang="ko" hreflang="ko" data-title="원주율" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Pi" title="Pi — hausa" lang="ha" hreflang="ha" data-title="Pi" data-language-autonym="Hausa" data-language-local-name="hausa" 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) — havaiji" lang="haw" hreflang="haw" data-title="Pai (makemakika)" data-language-autonym="Hawaiʻi" data-language-local-name="havaiji" class="interlanguage-link-target"><span>Hawaiʻi</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 — armenia" lang="hy" hreflang="hy" data-title="Պի թիվ" data-language-autonym="Հայերեն" data-language-local-name="armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%88" title="पाई — hindi" lang="hi" hreflang="hi" data-title="पाई" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</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 — yläsorbi" lang="hsb" hreflang="hsb" data-title="Konstanta π" data-language-autonym="Hornjoserbsce" data-language-local-name="yläsorbi" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pi_(broj)" title="Pi (broj) — kroatia" lang="hr" hreflang="hr" data-title="Pi (broj)" data-language-autonym="Hrvatski" data-language-local-name="kroatia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/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-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-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи — osseetti" lang="os" hreflang="os" data-title="Пи" data-language-autonym="Ирон" data-language-local-name="osseetti" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Phi" title="Phi — xhosa" lang="xh" hreflang="xh" data-title="Phi" data-language-autonym="IsiXhosa" data-language-local-name="xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/P%C3%AD" title="Pí — islanti" lang="is" hreflang="is" data-title="Pí" data-language-autonym="Íslenska" data-language-local-name="islanti" 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 — italia" lang="it" hreflang="it" data-title="Pi greco" data-language-autonym="Italiano" data-language-local-name="italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%90%D7%99" title="פאי — heprea" lang="he" hreflang="he" data-title="פאי" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pi" title="Pi — jaava" lang="jv" hreflang="jv" data-title="Pi" data-language-autonym="Jawa" data-language-local-name="jaava" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%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-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) — georgia" lang="ka" hreflang="ka" data-title="პი (რიცხვი)" data-language-autonym="ქართული" data-language-local-name="georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%81%D0%B0%D0%BD)" title="Пи (сан) — kazakki" lang="kk" hreflang="kk" data-title="Пи (сан)" data-language-autonym="Қазақша" data-language-local-name="kazakki" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Pi" title="Pi — korni" lang="kw" hreflang="kw" data-title="Pi" data-language-autonym="Kernowek" data-language-local-name="korni" 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 — kirgiisi" lang="ky" hreflang="ky" data-title="Пи" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target"><span>Кыргызча</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-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Pi_(matematik)" title="Pi (matematik) — haiti" lang="ht" hreflang="ht" data-title="Pi (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haiti" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Pi" title="Pi — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Pi" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Pi" title="Pi — kurdi" lang="ku" hreflang="ku" data-title="Pi" data-language-autonym="Kurdî" data-language-local-name="kurdi" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://la.wikipedia.org/wiki/Numerus_pi" title="Numerus pi — latina" lang="la" hreflang="la" data-title="Numerus pi" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/P%C4%AB" title="Pī — latvia" lang="lv" hreflang="lv" data-title="Pī" data-language-autonym="Latviešu" data-language-local-name="latvia" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Pi_(Zuel)" title="Pi (Zuel) — luxemburg" lang="lb" hreflang="lb" data-title="Pi (Zuel)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburg" 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) — lezgi" lang="lez" hreflang="lez" data-title="Пи (число)" data-language-autonym="Лезги" data-language-local-name="lezgi" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pi" title="Pi — liettua" lang="lt" hreflang="lt" data-title="Pi" data-language-autonym="Lietuvių" data-language-local-name="liettua" class="interlanguage-link-target"><span>Lietuvių</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) — limburg" lang="li" hreflang="li" data-title="Pi (wiskónde)" data-language-autonym="Limburgs" data-language-local-name="limburg" 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 — lombardi" lang="lmo" hreflang="lmo" data-title="Pi gregh" data-language-autonym="Lombard" data-language-local-name="lombardi" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pi_(sz%C3%A1m)" title="Pi (szám) — unkari" lang="hu" hreflang="hu" data-title="Pi (szám)" data-language-autonym="Magyar" data-language-local-name="unkari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи — makedonia" lang="mk" hreflang="mk" data-title="Пи" data-language-autonym="Македонски" data-language-local-name="makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Pi" title="Pi — malagassi" lang="mg" hreflang="mg" data-title="Pi" data-language-autonym="Malagasy" data-language-local-name="malagassi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%88_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="പൈ (ഗണിതം) — malajalam" lang="ml" hreflang="ml" data-title="പൈ (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="malajalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%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-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) — egyptinarabia" lang="arz" hreflang="arz" data-title="باى (رياضيات)" data-language-autonym="مصرى" data-language-local-name="egyptinarabia" class="interlanguage-link-target"><span>مصرى</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-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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи — mongoli" lang="mn" hreflang="mn" data-title="Пи" data-language-autonym="Монгол" data-language-local-name="mongoli" class="interlanguage-link-target"><span>Монгол</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) — burma" lang="my" hreflang="my" data-title="ပိုင် (သင်္ချာ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Pi" title="Pi — fidži" lang="fj" hreflang="fj" data-title="Pi" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fidži" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) — hollanti" lang="nl" hreflang="nl" data-title="Pi (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="hollanti" class="interlanguage-link-target"><span>Nederlands</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 — nepali" lang="ne" hreflang="ne" data-title="पाई" data-language-autonym="नेपाली" data-language-local-name="nepali" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%86%E5%91%A8%E7%8E%87" title="円周率 — japani" lang="ja" hreflang="ja" data-title="円周率" data-language-autonym="日本語" data-language-local-name="japani" class="interlanguage-link-target"><span>日本語</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 — pohjoisfriisi" lang="frr" hreflang="frr" data-title="Pi" data-language-autonym="Nordfriisk" data-language-local-name="pohjoisfriisi" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Pi" title="Pi — norjan bokmål" lang="nb" hreflang="nb" data-title="Pi" data-language-autonym="Norsk bokmål" data-language-local-name="norjan bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Pi" title="Pi — norjan nynorsk" lang="nn" hreflang="nn" data-title="Pi" data-language-autonym="Norsk nynorsk" data-language-local-name="norjan nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Pi" title="Pi — oksitaani" lang="oc" hreflang="oc" data-title="Pi" data-language-autonym="Occitan" data-language-local-name="oksitaani" class="interlanguage-link-target"><span>Occitan</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 — orija" lang="or" hreflang="or" data-title="ପାଇ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="orija" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Pi" title="Pi — uzbekki" lang="uz" hreflang="uz" data-title="Pi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 — pandžabi" lang="pa" hreflang="pa" data-title="ਪਾਈ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl — pfaltsi" lang="pfl" hreflang="pfl" data-title="Kreiszahl" data-language-autonym="Pälzisch" data-language-local-name="pfaltsi" class="interlanguage-link-target"><span>Pälzisch</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 ) — paštu" lang="ps" hreflang="ps" data-title="پای ( فزيک )" data-language-autonym="پښتو" data-language-local-name="paštu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Pi" title="Pi — jamaikankreolienglanti" lang="jam" hreflang="jam" data-title="Pi" data-language-autonym="Patois" data-language-local-name="jamaikankreolienglanti" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Pi_(nombe)" title="Pi (nombe) — picardi" lang="pcd" hreflang="pcd" data-title="Pi (nombe)" data-language-autonym="Picard" data-language-local-name="picardi" class="interlanguage-link-target"><span>Picard</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 — piemonte" lang="pms" hreflang="pms" data-title="Nùmer ëd Ludolph" data-language-autonym="Piemontèis" data-language-local-name="piemonte" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krinktall" title="Krinktall — alasaksa" lang="nds" hreflang="nds" data-title="Krinktall" data-language-autonym="Plattdüütsch" data-language-local-name="alasaksa" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pi" title="Pi — puola" lang="pl" hreflang="pl" data-title="Pi" data-language-autonym="Polski" data-language-local-name="puola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://pt.wikipedia.org/wiki/Pi" title="Pi — portugali" lang="pt" hreflang="pt" data-title="Pi" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</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-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://ro.wikipedia.org/wiki/Pi" title="Pi — romania" lang="ro" hreflang="ro" data-title="Pi" data-language-autonym="Română" data-language-local-name="romania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chiqaluwa" title="Chiqaluwa — ketšua" lang="qu" hreflang="qu" data-title="Chiqaluwa" data-language-autonym="Runa Simi" data-language-local-name="ketšua" class="interlanguage-link-target"><span>Runa Simi</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 — ruteeni" lang="rue" hreflang="rue" data-title="Чісло пі" data-language-autonym="Русиньскый" data-language-local-name="ruteeni" class="interlanguage-link-target"><span>Русиньскый</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) — venäjä" lang="ru" hreflang="ru" data-title="Пи (число)" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи — jakuutti" lang="sah" hreflang="sah" data-title="Пи" data-language-autonym="Саха тыла" data-language-local-name="jakuutti" class="interlanguage-link-target"><span>Саха тыла</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 — sanskrit" lang="sa" hreflang="sa" data-title="प्या" data-language-autonym="संस्कृतम्" data-language-local-name="sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</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 — sardi" lang="sc" hreflang="sc" data-title="Pi grecu" data-language-autonym="Sardu" data-language-local-name="sardi" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Pi" title="Pi — skotti" lang="sco" hreflang="sco" data-title="Pi" data-language-autonym="Scots" data-language-local-name="skotti" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri_pi" title="Numri pi — albania" lang="sq" hreflang="sq" data-title="Numri pi" data-language-autonym="Shqip" data-language-local-name="albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Pi_grecu" title="Pi grecu — sisilia" lang="scn" hreflang="scn" data-title="Pi grecu" data-language-autonym="Sicilianu" data-language-local-name="sisilia" class="interlanguage-link-target"><span>Sicilianu</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) — sinhala" lang="si" hreflang="si" data-title="පයි (අංකය)" data-language-autonym="සිංහල" data-language-local-name="sinhala" 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 — slovakki" lang="sk" hreflang="sk" data-title="Ludolfovo číslo" data-language-autonym="Slovenčina" data-language-local-name="slovakki" 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 — sloveeni" lang="sl" hreflang="sl" data-title="Pi" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Pi" title="Pi — sleesia" lang="szl" hreflang="szl" data-title="Pi" data-language-autonym="Ślůnski" data-language-local-name="sleesia" class="interlanguage-link-target"><span>Ślůnski</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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%D8%A7%DB%8C" title="پای — soranî" lang="ckb" hreflang="ckb" data-title="پای" data-language-autonym="کوردی" data-language-local-name="soranî" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи — serbia" lang="sr" hreflang="sr" data-title="Пи" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Pi" title="Pi — serbokroaatti" lang="sh" hreflang="sh" data-title="Pi" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaatti" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pi" title="Pi — ruotsi" lang="sv" hreflang="sv" data-title="Pi" data-language-autonym="Svenska" data-language-local-name="ruotsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Pi" title="Pi — tašelhit" lang="shi" hreflang="shi" data-title="Pi" data-language-autonym="Taclḥit" data-language-local-name="tašelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pi" title="Pi — tagalog" lang="tl" hreflang="tl" data-title="Pi" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%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) — tamili" lang="ta" hreflang="ta" data-title="பை (கணித மாறிலி)" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Pi" title="Pi — kabyyli" lang="kab" hreflang="kab" data-title="Pi" data-language-autonym="Taqbaylit" data-language-local-name="kabyyli" class="interlanguage-link-target"><span>Taqbaylit</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-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 — tataari" lang="tt" hreflang="tt" data-title="Пи саны" data-language-autonym="Татарча / tatarça" data-language-local-name="tataari" class="interlanguage-link-target"><span>Татарча / tatarça</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 — telugu" lang="te" hreflang="te" data-title="పై" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%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) — thai" lang="th" hreflang="th" data-title="พาย (ค่าคงตัว)" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://vi.wikipedia.org/wiki/Pi" title="Pi — vietnam" lang="vi" hreflang="vi" data-title="Pi" data-language-autonym="Tiếng Việt" data-language-local-name="vietnam" class="interlanguage-link-target"><span>Tiếng Việt</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) — tadžikki" lang="tg" hreflang="tg" data-title="Пӣ (адад)" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžikki" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Pi_say%C4%B1s%C4%B1" title="Pi sayısı — turkki" lang="tr" hreflang="tr" data-title="Pi sayısı" data-language-autonym="Türkçe" data-language-local-name="turkki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BF%D1%96" title="Число пі — ukraina" lang="uk" hreflang="uk" data-title="Число пі" data-language-autonym="Українська" data-language-local-name="ukraina" 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 — urdu" lang="ur" hreflang="ur" data-title="پائی" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Pi_greco" title="Pi greco — venetsia" lang="vec" hreflang="vec" data-title="Pi greco" data-language-autonym="Vèneto" data-language-local-name="venetsia" 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) — vepsä" lang="vep" hreflang="vep" data-title="Pi (lugu)" data-language-autonym="Vepsän kel’" data-language-local-name="vepsä" class="interlanguage-link-target"><span>Vepsän kel’</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-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 — klassinen kiina" lang="lzh" hreflang="lzh" data-title="圓周率" data-language-autonym="文言" data-language-local-name="klassinen kiina" class="interlanguage-link-target"><span>文言</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 — wu-kiina" lang="wuu" hreflang="wuu" data-title="圓周率" data-language-autonym="吴语" data-language-local-name="wu-kiina" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%99" title="פי — jiddiš" lang="yi" hreflang="yi" data-title="פי" data-language-autonym="ייִדיש" data-language-local-name="jiddiš" 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 — joruba" lang="yo" hreflang="yo" data-title="Pi" data-language-autonym="Yorùbá" data-language-local-name="joruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 — kantoninkiina" lang="yue" hreflang="yue" data-title="圓周率" data-language-autonym="粵語" data-language-local-name="kantoninkiina" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amar_Pi" title="Amar Pi — Dimli" lang="diq" hreflang="diq" data-title="Amar Pi" data-language-autonym="Zazaki" data-language-local-name="Dimli" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) — seelanti" lang="zea" hreflang="zea" data-title="Pi (wiskunde)" data-language-autonym="Zeêuws" data-language-local-name="seelanti" class="interlanguage-link-target"><span>Zeêuws</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Pi" title="Pi — samogiitti" lang="sgs" hreflang="sgs" data-title="Pi" data-language-autonym="Žemaitėška" data-language-local-name="samogiitti" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://zh.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 — kiina" lang="zh" hreflang="zh" data-title="圓周率" data-language-autonym="中文" data-language-local-name="kiina" class="interlanguage-link-target"><span>中文</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> </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="Muokkaa kieltenvälisiä linkkejä" class="wbc-editpage">Muokkaa linkkejä</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="Nimiavaruudet"> <div 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src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/250px-Pi-unrolled-720.gif" decoding="async" width="250" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/375px-Pi-unrolled-720.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/500px-Pi-unrolled-720.gif 2x" data-file-width="720" data-file-height="228" /></a><figcaption>Kun <a href="/wiki/Ympyr%C3%A4" title="Ympyrä">ympyrän</a> <a href="/wiki/Halkaisija" title="Halkaisija">halkaisija</a> on <a href="/wiki/1_(luku)" title="1 (luku)">1</a>, ympyrän <a href="/wiki/Keh%C3%A4_(geometria)" title="Kehä (geometria)">kehä</a> on pii.</figcaption></figure> <p><b>Pii</b> eli <a href="/wiki/Pii_(kirjain)" title="Pii (kirjain)"><b>π</b></a> on matemaattinen <a href="/wiki/Vakio" title="Vakio">vakio</a>, <a href="/wiki/Ympyr%C3%A4" title="Ympyrä">ympyrän</a> kehän suhde halkaisijaan <a href="/wiki/Geometria" title="Geometria">euklidisessa geometriassa</a>. Pii esiintyy monilla <a href="/wiki/Matematiikka" title="Matematiikka">matematiikan</a> ja <a href="/wiki/Fysiikka" title="Fysiikka">fysiikan</a> alueilla. </p><p>Vaihtoehtoisesti pii voidaan määritellä <i>r</i>-säteisen ympyrän pinta-alan suhteena <i>r</i>-sivuisen neliön pinta-alaan: <span class="mwe-math-element"><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 r^{2}}{r^{2}}}=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi r^{2}}{r^{2}}}=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4548afb31d80319e84373ebb9c2d3983ff193291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.701ex; height:6.009ex;" alt="{\displaystyle {\frac {\pi r^{2}}{r^{2}}}=\pi }" /></span>. Joissain analyysin kirjoissa pii määritellään pienimmäksi positiiviseksi luvuksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>, jolle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66eaafb8fb45867ab892617867afaed4a8ac49f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:2.843ex;" alt="{\displaystyle \sin(x)=0}" /></span>. </p><p>Piin likiarvo katkaistuna 100 desimaalin jälkeen on 3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679. </p><p><a href="/wiki/Eukleides" title="Eukleides">Eukleideen</a> <a href="/wiki/Alkeet" title="Alkeet">Alkeet</a>-teoksen luvussa XII todistetaan, että kahden ympyrän alan suhde on sama kuin niiden halkaisijoiden <a href="/wiki/Neli%C3%B6_(algebra)" title="Neliö (algebra)">neliöiden</a> suhde. Tästä seuraa, että ympyrän pinta-ala on vakio (= π / 4) kertaa sen halkaisijan neliö. Pii on <a href="/wiki/Irrationaaliluku" title="Irrationaaliluku">irrationaaliluku</a> eli luku, jonka <a href="/wiki/Desimaaliluku" title="Desimaaliluku">desimaalikehitelmä</a> on päättymätön ja <a href="/wiki/Jaksollinen_desimaaliluku" title="Jaksollinen desimaaliluku">jaksoton</a>. <a href="/wiki/Ferdinand_Lindemann" class="mw-redirect" title="Ferdinand Lindemann">Ferdinand Lindemann</a> todisti vuonna 1882 piin olevan <a href="/wiki/Transsendenttiluku" title="Transsendenttiluku">transsendenttiluku</a>, eli luku, joka ei ole minkään rationaalilukukertoimisen <a href="/wiki/Polynomi" title="Polynomi">polynomin</a> nollakohta. </p><p>Joissakin maissa pii tunnetaan myös nimillä <i>Arkhimedeen vakio</i><sup id="cite_ref-archimedes_1-0" class="reference"><a href="#cite_note-archimedes-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ja (erityisesti saksankielisellä alueella) <i>Ludolphin luku</i><sup id="cite_ref-ludolph_2-0" class="reference"><a href="#cite_note-ludolph-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> matemaatikkojen <a href="/wiki/Arkhimedes" title="Arkhimedes">Arkhimedes</a> ja <a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Ludolph van Ceulen</a> mukaan. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Piin_vanhoja_likiarvoja">Piin vanhoja likiarvoja</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=1" title="Muokkaa osiota Piin vanhoja likiarvoja" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=1" title="Muokkaa osion lähdekoodia: Piin vanhoja likiarvoja"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Tiedosto:Greek_lc_pi_icon.svg" class="mw-file-description" title="Pii"><img alt="Pii" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Greek_lc_pi_icon.svg/120px-Greek_lc_pi_icon.svg.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Greek_lc_pi_icon.svg/180px-Greek_lc_pi_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Greek_lc_pi_icon.svg/240px-Greek_lc_pi_icon.svg.png 2x" data-file-width="100" data-file-height="96" /></a><figcaption>Pii</figcaption></figure> <p>Koska pii on <a href="/wiki/Transsendenttiluku" title="Transsendenttiluku">transsendenttiluku</a>, sitä ei voi esittää päättyvänä lausekkeena <a href="/wiki/Peruslaskutoimitukset" class="mw-redirect" title="Peruslaskutoimitukset">peruslaskutoimituksia</a>, <a href="/wiki/Potenssi" title="Potenssi">potenssiinkorotusta</a> ja <a href="/wiki/Neli%C3%B6juuri" title="Neliöjuuri">juurenottoa</a> käyttäen. Sitä on kuitenkin kauan arvioitu likimääräisesti. <a href="/wiki/Vanha_testamentti" title="Vanha testamentti">Vanhan testamentin</a> <a href="/wiki/Ensimm%C3%A4inen_kuninkaiden_kirja" title="Ensimmäinen kuninkaiden kirja">Ensimmäisessä kuninkaiden kirjassa</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> on 3: ”Hiram valoi myös pyöreän altaan, jota kutsuttiin mereksi. Se oli reunasta reunaan kymmenen <a href="/wiki/Kyyn%C3%A4r%C3%A4" title="Kyynärä">kyynärän</a> levyinen, korkeutta sillä oli viisi kyynärää, ja vasta kolmenkymmenen kyynärän pituinen mittanuora ulottui sen ympäri”.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Ensimmäisiä säällisiä säilyneitä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>:n likiarvoja on <a href="/wiki/Egypti" title="Egypti">egyptiläisen</a> matemaatikko <a href="/w/index.php?title=Ahmose_(matemaatikko)&action=edit&redlink=1" class="new" title="Ahmose (matemaatikko) (sivua ei ole)">Ahmosen</a> käyttämä. Se on säilynyt laskutehtävissä, jotka sisältyvät niin sanottuun <a href="/wiki/Rhindin_papyrus" title="Rhindin papyrus">Rhindin papyrukseen</a>. Sen mukaan ympyrän pinta-ala on yhtä suuri kuin sellaisen neliön, jonka sivu on 8/9 ympyrän halkaisijasta. Tämä vastaa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>:n likiarvoa 256/81 eli noin 3,16. Noin 2 000 vuotta ennen ajanlaskun alkua <a href="/wiki/Babylonialaiset" class="mw-redirect" title="Babylonialaiset">babylonialaiset</a> otaksuivat, että <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> on joko <b>3</b> tai <span class="mwe-math-element"><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 {25}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {25}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5374f1d2966a9af3579f84900a272b125302ba03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {25}{8}}}" /></span> (yksi desimaali oikein). Myös likiarvo <span class="mwe-math-element"><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 {22}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>22</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {22}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a48b7aaf32f5f27ba3ffc352f16e3e0ed95680e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.161ex; height:5.343ex;" alt="{\displaystyle {\frac {22}{7}}}" /></span> (kaksi des. oikein) on tiedetty pitkään. </p><p><a href="/wiki/Antiikin_Kreikka" title="Antiikin Kreikka">Kreikkalainen</a> filosofi ja matemaatikko <a href="/wiki/Arkhimedes" title="Arkhimedes">Arkhimedes</a> todisti ympyrän sisään ja ympärille piirrettyjen <a href="/wiki/Monikulmio" title="Monikulmio">monikulmioiden</a> avulla, että ympyrän kehän ja halkaisijan suhde on lukujen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3{\frac {1}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{\frac {1}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd1e13460c1ade1ceb0bf018020ec5cc069ab2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.161ex; height:5.343ex;" alt="{\displaystyle 3{\frac {1}{7}}}" /></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3{\frac {10}{71}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>71</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{\frac {10}{71}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ee5c133166cd8fd4ed512d8b8c44bdf4a78145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.323ex; height:5.343ex;" alt="{\displaystyle 3{\frac {10}{71}}}" /></span> välillä.<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> <a href="/wiki/Klaudios_Ptolemaios" title="Klaudios Ptolemaios">Ptolemaios</a> käytti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>:n arvoa <span class="mwe-math-element"><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 {377}{120}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>377</mn> <mn>120</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {377}{120}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03952bdd28084d07e8e5cb39f387efc9fa343d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.323ex; height:5.176ex;" alt="{\displaystyle {\frac {377}{120}}}" /></span> (kolme des. oikein). <a href="/wiki/Kiinalaiset" class="mw-redirect" title="Kiinalaiset">Kiinalainen</a> <a href="/w/index.php?title=Tsi_Ch%27ung-Chi&action=edit&redlink=1" class="new" title="Tsi Ch'ung-Chi (sivua ei ole)">Tsi Ch'ung-Chi</a> löysi 400-luvulla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>:lle arvon <span class="mwe-math-element"><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 {355}{113}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>355</mn> <mn>113</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {355}{113}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee93c242d2b1cbb29fe7d0a0b9e4142328ce3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.323ex; height:5.176ex;" alt="{\displaystyle {\frac {355}{113}}}" /></span> (kuusi des. oikein), jota parempi murtolukuarvio on vasta <span class="mwe-math-element"><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 {103993}{33102}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>103993</mn> <mn>33102</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {103993}{33102}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdafe8a9967f79acf48ad7a3d328ee2cb781e181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.811ex; height:5.176ex;" alt="{\displaystyle {\frac {103993}{33102}}}" /></span> (yhdeksän des. oikein). </p><p>Luku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> todistettiin <a href="/wiki/Irrationaaliluku" title="Irrationaaliluku">irrationaaliluvuksi</a> 1700-luvulla. </p> <div class="mw-heading mw-heading2"><h2 id="Piin_laskeminen_sarjakehitelmien_avulla">Piin laskeminen sarjakehitelmien avulla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=2" title="Muokkaa osiota Piin laskeminen sarjakehitelmien avulla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=2" title="Muokkaa osion lähdekoodia: Piin laskeminen sarjakehitelmien avulla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Piin voi esittää päättymättömänä <a href="/wiki/Sarja_(matematiikka)" title="Sarja (matematiikka)">sarjana</a>. Eräs varhainen ja yksinkertainen tapa määritellä pii sarjana on <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibnizin</a> kehittämä <a href="/w/index.php?title=Gregory%E2%80%93Leibniz-sarja&action=edit&redlink=1" class="new" title="Gregory–Leibniz-sarja (sivua ei ole)">Gregory–Leibniz-sarja</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 {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>11</mn> </mfrac> </mrow> <mo>⋯</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f849cd95a6dabc4f27e711ad2623cb26822fae01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.284ex; height:5.343ex;" alt="{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots \,}" /></span></dd></dl> <p>Tämä sarja <a href="/wiki/Suppeneminen" title="Suppeneminen">suppenee</a> kuitenkin liian hitaasti, jotta sitä kannattaisi käyttää piin likiarvojen laskemiseen. Siitä olisi laskettava vähintään 294 ensimmäistä termiä, jotta saataisiin edes kaksidesimaalinen likiarvo 3,14. Vuonna 1706 <a href="/wiki/John_Machin" title="John Machin">John Machin</a> todisti kuitenkin seuraavan yhtälön: </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 }{4}}=4\,\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>239</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=4\,\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5556a4c83b654a249fef903f209c18fc1c2755cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.108ex; width:29.793ex; height:5.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=4\,\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}\!}" /></span></dd></dl> <p>Koska <a href="/wiki/Arkustangentti" class="mw-redirect" title="Arkustangentti">arkustangentin</a> <a href="/wiki/Taylorin_sarja" title="Taylorin sarja">Taylorin sarjakehitelmä</a> on </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 \arctan \,x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan \,x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dabbe6d1739cef0bde2b5a9b76d112ab312a6b56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.204ex; width:36.56ex; height:5.843ex;" alt="{\displaystyle \arctan \,x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \!}" /></span></dd></dl> <p>saatiin tästä piille nopeasti suppeneva ja käyttökelpoinen sarjakehitelmä: </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 }{4}}={\frac {4}{5}}-{\frac {4}{3\cdot 5^{3}}}+{\frac {4}{5\cdot 5^{5}}}-{\frac {4}{7\cdot 5^{7}}}+....-{\frac {1}{239}}+{\frac {1}{3\cdot 239^{3}}}-{\frac {1}{5\cdot 239^{5}}}+{\frac {1}{7\cdot 239^{7}}}-...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>5</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>7</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>239</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <msup> <mn>239</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo>⋅</mo> <msup> <mn>239</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>7</mn> <mo>⋅</mo> <msup> <mn>239</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}={\frac {4}{5}}-{\frac {4}{3\cdot 5^{3}}}+{\frac {4}{5\cdot 5^{5}}}-{\frac {4}{7\cdot 5^{7}}}+....-{\frac {1}{239}}+{\frac {1}{3\cdot 239^{3}}}-{\frac {1}{5\cdot 239^{5}}}+{\frac {1}{7\cdot 239^{7}}}-...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e281a6ca2b3a482d36b7bd15904dc0306746d196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:83.247ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi }{4}}={\frac {4}{5}}-{\frac {4}{3\cdot 5^{3}}}+{\frac {4}{5\cdot 5^{5}}}-{\frac {4}{7\cdot 5^{7}}}+....-{\frac {1}{239}}+{\frac {1}{3\cdot 239^{3}}}-{\frac {1}{5\cdot 239^{5}}}+{\frac {1}{7\cdot 239^{7}}}-...}" /></span></dd></dl> <p>Machin itse laski tällä kaavalla piin 100 desimaalin tarkkuudella, ja myöhemminkin tätä sarjaa on paljon käytetty yhä tarkempien likiarvojen laskemiseen. </p><p><a href="/wiki/Alkuluku" title="Alkuluku">Alkulukujen</a> 7, 11, 13, … avulla on johdettu tulokaava<sup id="cite_ref-JM_5-0" class="reference"><a href="#cite_note-JM-5"><span class="cite-bracket">[</span>5<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 =[945({7^{6} \over 7^{6}-1}\cdot {11^{6} \over 11^{6}-1}\cdot {13^{6} \over 13^{6}-1}\cdot ...)]^{1 \over 6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>945</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <msup> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =[945({7^{6} \over 7^{6}-1}\cdot {11^{6} \over 11^{6}-1}\cdot {13^{6} \over 13^{6}-1}\cdot ...)]^{1 \over 6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673c5ca2bff577084ed9e0ec3f37168ad0f8593e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.355ex; height:6.343ex;" alt="{\displaystyle \pi =[945({7^{6} \over 7^{6}-1}\cdot {11^{6} \over 11^{6}-1}\cdot {13^{6} \over 13^{6}-1}\cdot ...)]^{1 \over 6}}" /></span></dd></dl> <p>Edellistä pienemmistä alkuluvuista 2, 3, 5, … lähtien pätee myös tulokaava<sup id="cite_ref-JM_5-1" class="reference"><a href="#cite_note-JM-5"><span class="cite-bracket">[</span>5<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 (1-{1 \over 2^{2}})(1-{1 \over 3^{2}})(1-{1 \over 5^{2}})...={6 \over \pi ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <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 stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <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 stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-{1 \over 2^{2}})(1-{1 \over 3^{2}})(1-{1 \over 5^{2}})...={6 \over \pi ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6188415fceef78d67adf564be902dc9be5f0eed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.375ex; height:5.676ex;" alt="{\displaystyle (1-{1 \over 2^{2}})(1-{1 \over 3^{2}})(1-{1 \over 5^{2}})...={6 \over \pi ^{2}}}" /></span></dd></dl> <p>Vuorottelevista sarjoista voidaan mainita tulos<sup id="cite_ref-JM_5-2" class="reference"><a href="#cite_note-JM-5"><span class="cite-bracket">[</span>5<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 \sum _{k=1}^{\infty }{(-1)^{k-1} \over k^{2}}={\pi ^{2} \over 12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{(-1)^{k-1} \over k^{2}}={\pi ^{2} \over 12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147d3683d802cec730bb25b23dabac130f3fb047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.87ex; height:7.009ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{(-1)^{k-1} \over k^{2}}={\pi ^{2} \over 12}}" /></span>,</dd></dl> <p>josta saadaan </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 ={\sqrt {12(1-{1 \over 4}+{1 \over 9}-{1 \over 16}+{1 \over 25}-...)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>25</mn> </mfrac> </mrow> <mo>−</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\sqrt {12(1-{1 \over 4}+{1 \over 9}-{1 \over 16}+{1 \over 25}-...)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08a2f316668721e8e62f8d72b093c4f96054dec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.642ex; height:6.176ex;" alt="{\displaystyle \pi ={\sqrt {12(1-{1 \over 4}+{1 \over 9}-{1 \over 16}+{1 \over 25}-...)}}}" /></span></dd></dl> <p>Muita äärettömiä sarjoja: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>k</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <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> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>k</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425b657215f33eb7b5e63abe69d5a1fb70eceda2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.108ex; width:35.555ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}\!}" /></span></dd></dl> <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 {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)=\pi \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </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> <mo stretchy="false">(</mo> <mn>4</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>6</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>12</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>127169</mn> <mrow> <mn>12</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1070</mn> <mrow> <mn>12</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>131</mn> <mrow> <mn>12</mn> <mi>k</mi> <mo>+</mo> <mn>7</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>12</mn> <mi>k</mi> <mo>+</mo> <mn>11</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>π</mi> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)=\pi \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e04c1dc8c543027ffaabce1f40fd3b54f48c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.387ex; width:75.496ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)=\pi \!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)=\pi \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mrow> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>7</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>π</mi> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)=\pi \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3df7f12b0b5b1217a0d9d396064b605060eb4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:98.691ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)=\pi \!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28428bc91179e740b3ef941846cdcbb13f2733bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:24.082ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi \!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93cfdfe29db4e5d5ed69ecc87ab5fbb0616416eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:61.732ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}\!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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>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>5</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>7</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> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9c039251c9444e57f049c637308f6f3dc6670e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:51.053ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}\!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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>3</mn> </mrow> </msup> <mn>32</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be5448dcfa4f9e12e19bf14b714a1672088a09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:51.053ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}\!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <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>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>5</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>7</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> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>96</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745c65e60136158f32846b3bfb17ec157ca35603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:51.053ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}\!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</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>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>1536</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd0f092621d78d0e20ae89741bea2292c53b9b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:53.314ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}\!}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</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>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</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>6</mn> </mrow> </msup> <mn>960</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f7c7e6b10f18cc5169ebd8ac6144c3aecee347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:52.151ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}\!}" /></span></dd></dl> <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 ={\sqrt {12}}\,\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{5\cdot 3^{2}}}-{\frac {1}{7\cdot 3^{3}}}+\cdots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>7</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\sqrt {12}}\,\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{5\cdot 3^{2}}}-{\frac {1}{7\cdot 3^{3}}}+\cdots \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc41d4b070b73933fb965d40b10e54a30f57c14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.763ex; height:6.176ex;" alt="{\displaystyle \pi ={\sqrt {12}}\,\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{5\cdot 3^{2}}}-{\frac {1}{7\cdot 3^{3}}}+\cdots \right)}" /></span></dd></dl> <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 =8\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {1}{(4m-2)^{2k}}}=4\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {m^{2}-k^{2}}{(m^{2}+k^{2})^{2}}}={\sqrt[{4\,\,}]{360\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{k}{\frac {1}{m(k+1)^{3}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>8</mn> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>m</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>360</mn> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =8\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {1}{(4m-2)^{2k}}}=4\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {m^{2}-k^{2}}{(m^{2}+k^{2})^{2}}}={\sqrt[{4\,\,}]{360\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{k}{\frac {1}{m(k+1)^{3}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637f3bd1f1f43c56cd7c73ea811a470a11734006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:77.78ex; height:8.009ex;" alt="{\displaystyle \pi =8\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {1}{(4m-2)^{2k}}}=4\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{\infty }{\frac {m^{2}-k^{2}}{(m^{2}+k^{2})^{2}}}={\sqrt[{4\,\,}]{360\sum \limits _{k=1}^{\infty }\sum \limits _{m=1}^{k}{\frac {1}{m(k+1)^{3}}}}}}" /></span>.</dd></dl> <p>Muutama BBP-kaava: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi &={\tfrac {1}{2}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}+{\tfrac {4}{8k+4}}-{\tfrac {1}{8k+7}}\right)\\&={\tfrac {1}{4}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+1}}+{\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}-{\tfrac {2}{8k+5}}-{\tfrac {2}{8k+6}}-{\tfrac {1}{8k+7}}\right)\\&=\;\;\sum _{k=0}^{\infty }{\tfrac {(-1)^{k}}{4^{k}}}\left({\tfrac {2}{4k+1}}+{\tfrac {2}{4k+2}}+{\tfrac {1}{4k+3}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </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"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>7</mn> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </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"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>7</mn> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <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"> <mstyle displaystyle="false" scriptlevel="0"> <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> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi &={\tfrac {1}{2}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}+{\tfrac {4}{8k+4}}-{\tfrac {1}{8k+7}}\right)\\&={\tfrac {1}{4}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+1}}+{\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}-{\tfrac {2}{8k+5}}-{\tfrac {2}{8k+6}}-{\tfrac {1}{8k+7}}\right)\\&=\;\;\sum _{k=0}^{\infty }{\tfrac {(-1)^{k}}{4^{k}}}\left({\tfrac {2}{4k+1}}+{\tfrac {2}{4k+2}}+{\tfrac {1}{4k+3}}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f688492f520d411a7ce8ce16e4618eb451a9d4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:59.364ex; height:21.343ex;" alt="{\displaystyle {\begin{aligned}\pi &={\tfrac {1}{2}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}+{\tfrac {4}{8k+4}}-{\tfrac {1}{8k+7}}\right)\\&={\tfrac {1}{4}}\sum _{k=0}^{\infty }{\tfrac {1}{16^{k}}}\left({\tfrac {8}{8k+1}}+{\tfrac {8}{8k+2}}+{\tfrac {4}{8k+3}}-{\tfrac {2}{8k+5}}-{\tfrac {2}{8k+6}}-{\tfrac {1}{8k+7}}\right)\\&=\;\;\sum _{k=0}^{\infty }{\tfrac {(-1)^{k}}{4^{k}}}\left({\tfrac {2}{4k+1}}+{\tfrac {2}{4k+2}}+{\tfrac {1}{4k+3}}\right).\end{aligned}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Piin_laskeminen_tulokehitelmien_avulla">Piin laskeminen tulokehitelmien avulla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=3" title="Muokkaa osiota Piin laskeminen tulokehitelmien avulla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=3" title="Muokkaa osion lähdekoodia: Piin laskeminen tulokehitelmien avulla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Wallisin_tulo" class="mw-redirect" title="Wallisin tulo">Wallisin tulo</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 \prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\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}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>15</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>36</mn> <mn>35</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>63</mn> </mfrac> </mrow> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\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}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63b917f506024ba7e2eabbb800cb081d691cfe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.108ex; width:73.591ex; height:6.843ex;" alt="{\displaystyle \prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\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}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}\!}" /></span></dd></dl> <p><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viètan</a> kaava: </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 {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c4cc19e4cfbb41dd3a1c43a4ec8326d7be65c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.108ex; width:43.532ex; height:9.009ex;" alt="{\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}\!}" /></span>.</dd></dl> <p>Muita äärettömiä tuloja: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =2{\sqrt {3}}\prod \limits _{k=1}^{\infty }{\frac {\left(2k-1\right)^{{\frac {1}{2}}-k}\left(2k+3\right)^{k+{\frac {1}{2}}}}{2k+1}}\left({\frac {k}{k+1}}\right)^{2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =2{\sqrt {3}}\prod \limits _{k=1}^{\infty }{\frac {\left(2k-1\right)^{{\frac {1}{2}}-k}\left(2k+3\right)^{k+{\frac {1}{2}}}}{2k+1}}\left({\frac {k}{k+1}}\right)^{2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc9a623ef699864729f880d2c28049b4e69e059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.766ex; height:8.009ex;" alt="{\displaystyle \pi =2{\sqrt {3}}\prod \limits _{k=1}^{\infty }{\frac {\left(2k-1\right)^{{\frac {1}{2}}-k}\left(2k+3\right)^{k+{\frac {1}{2}}}}{2k+1}}\left({\frac {k}{k+1}}\right)^{2k}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{e}}=2\prod \limits _{k=1}^{\infty }\left({\frac {2k+1}{2k-1}}\right)^{2k-1}\left({\frac {k}{k+1}}\right)^{2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>e</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{e}}=2\prod \limits _{k=1}^{\infty }\left({\frac {2k+1}{2k-1}}\right)^{2k-1}\left({\frac {k}{k+1}}\right)^{2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7d3480334a849c1b9ad42ef8ed443385f47a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.2ex; height:7.176ex;" alt="{\displaystyle {\frac {\pi }{e}}=2\prod \limits _{k=1}^{\infty }\left({\frac {2k+1}{2k-1}}\right)^{2k-1}\left({\frac {k}{k+1}}\right)^{2k}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \cdot e=6\prod \limits _{k=1}^{\infty }\left({\frac {2k+3}{2k+1}}\right)^{2k+1}\left({\frac {k}{k+1}}\right)^{2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>⋅</mo> <mi>e</mi> <mo>=</mo> <mn>6</mn> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \cdot e=6\prod \limits _{k=1}^{\infty }\left({\frac {2k+3}{2k+1}}\right)^{2k+1}\left({\frac {k}{k+1}}\right)^{2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a4ecd484d91eb5b29724a2437938ac6db2fa21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.126ex; height:7.176ex;" alt="{\displaystyle \pi \cdot e=6\prod \limits _{k=1}^{\infty }\left({\frac {2k+3}{2k+1}}\right)^{2k+1}\left({\frac {k}{k+1}}\right)^{2k}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {3}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {2}{3}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {3}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {2}{3}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bd652e0bc206873ad863e5b939f74f56cda91b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:27.216ex; height:9.009ex;" alt="{\displaystyle \pi ={\frac {3}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {2}{3}}\right)^{2}}}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {4}{3}}\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {3}{4}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {4}{3}}\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {3}{4}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130abfe1f0c2b0845a609581a6d1d67dcc3c81f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:27.216ex; height:9.009ex;" alt="{\displaystyle \pi ={\frac {4}{3}}\cdot {\frac {\sqrt {2}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {3}{4}}\right)^{2}}}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {6}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {5}{6}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <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>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {6}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {5}{6}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815990ad9a54117c6dca41b492ff6c5695d56693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:25.28ex; height:8.843ex;" alt="{\displaystyle \pi ={\frac {6}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}}{k^{2}-\left({\frac {5}{6}}\right)^{2}}}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {9}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {2}{9}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>9</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {9}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {2}{9}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc46aa331c4ccab716bbdf621539c6759df7a931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.438ex; height:7.509ex;" alt="{\displaystyle \pi ={\frac {9}{2}}\cdot {\frac {\sqrt {3}}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {2}{9}}}}}" /></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {36}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {5}{36}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>36</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>36</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {36}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {5}{36}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b187e3ea89596d0b2eb26b449a3d1d0afe8b7585" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:28.133ex; height:7.343ex;" alt="{\displaystyle \pi ={\frac {36}{5}}\cdot {\frac {1}{2}}\prod \limits _{k=1}^{\infty }{\frac {k^{2}+k}{k^{2}+k+{\frac {5}{36}}}}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Piin_approksimaatioita">Piin approksimaatioita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=4" title="Muokkaa osiota Piin approksimaatioita" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=4" title="Muokkaa osion lähdekoodia: Piin approksimaatioita"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Piin avulla laskiessa voidaan lausekkeissa korvata se erityisellä lausekkeella. </p> <ul><li>kaksi oikeaa desimaalia:</li></ul> <dl><dd><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 {\sqrt {2}}+{\sqrt {3}}=3{,}146^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mn>3,146</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}=3{,}146^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a0aa2076e704173946df2df9ae8a333b9c4370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.943ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}=3{,}146^{+}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {15}}-{\sqrt {3}}+1=3{,}140^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mn>3,140</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {15}}-{\sqrt {3}}+1=3{,}140^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f42e5eb711067beb947d89e8e6f695470ccd896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.108ex; height:2.843ex;" alt="{\displaystyle {\sqrt {15}}-{\sqrt {3}}+1=3{,}140^{+}}" /></span></dd></dl></dd></dl> <ul><li>kolme oikeaa desimaalia:</li></ul> <dl><dd><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 {\sqrt[{3}]{31}}=3{,}1413^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>31</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mn>3,141</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{31}}=3{,}1413^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39973f9b044aeff301037b2ee52829603968240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.329ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{31}}=3{,}1413^{+}}" /></span></dd></dl></dd></dl> <ul><li>kolme oikeaa desimaalia:</li></ul> <dl><dd><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 {\sqrt {7+{\sqrt {6+{\sqrt {5}}}}}}=3{,}1416^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mo>=</mo> <mn>3,141</mn> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {7+{\sqrt {6+{\sqrt {5}}}}}}=3{,}1416^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b260ac2036ed5b80577393520d80b5f3c87a228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.82ex; height:6.176ex;" alt="{\displaystyle {\sqrt {7+{\sqrt {6+{\sqrt {5}}}}}}=3{,}1416^{+}}" /></span><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></dd></dl></dd></dl> <ul><li><a href="/wiki/Ramanujan" class="mw-redirect" title="Ramanujan">Ramanujanin</a> kehittämä approksimaatio, kolme oikeaa desimaalia:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {9}{5}}+{\sqrt {\frac {9}{5}}}=3{,}1416^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>9</mn> <mn>5</mn> </mfrac> </msqrt> </mrow> <mo>=</mo> <mn>3,141</mn> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {9}{5}}+{\sqrt {\frac {9}{5}}}=3{,}1416^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5730d0151f1aa10affa4fd8f7aa45dd358871b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.23ex; height:6.176ex;" alt="{\displaystyle {\frac {9}{5}}+{\sqrt {\frac {9}{5}}}=3{,}1416^{+}}" /></span></dd></dl></dd></dl> <ul><li>neljä oikeaa desimaalia:</li></ul> <dl><dd><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 {7^{7}}{4^{9}}}=3{,}14156^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>3,141</mn> <msup> <mn>56</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7^{7}}{4^{9}}}=3{,}14156^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3f48e85860bd05e71944849afe381194e9a16e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.284ex; height:6.176ex;" alt="{\displaystyle {\frac {7^{7}}{4^{9}}}=3{,}14156^{+}}" /></span></dd></dl></dd></dl> <ul><li>kuusi oikeaa desimaalia:</li></ul> <dl><dd><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 {355}{113}}=3{,}14159\ 29^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>355</mn> <mn>113</mn> </mfrac> </mrow> <mo>=</mo> <mn>3,141</mn> <mn>59</mn> <mtext> </mtext> <msup> <mn>29</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {355}{113}}=3{,}14159\ 29^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5edebcf39d6fc9f448d31c0949a43f7d8019a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.46ex; height:5.176ex;" alt="{\displaystyle {\frac {355}{113}}=3{,}14159\ 29^{+}}" /></span></dd></dl></dd></dl> <ul><li>Ramanujanin kehittämä approksimaatio, kahdeksan oikeaa desimaalia:</li></ul> <dl><dd><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 {\sqrt[{4}]{3^{4}+2^{4}+{\frac {1}{2+({\frac {2}{3}})^{2}}}}}={\sqrt[{4}]{\frac {2143}{22}}}=3{,}14159\ 2652^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mn>2143</mn> <mn>22</mn> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mn>3,141</mn> <mn>59</mn> <mtext> </mtext> <msup> <mn>2652</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{3^{4}+2^{4}+{\frac {1}{2+({\frac {2}{3}})^{2}}}}}={\sqrt[{4}]{\frac {2143}{22}}}=3{,}14159\ 2652^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6253386cd7e8a06b3f5d6feffa8c4e52c5e30941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:50.168ex; height:7.676ex;" alt="{\displaystyle {\sqrt[{4}]{3^{4}+2^{4}+{\frac {1}{2+({\frac {2}{3}})^{2}}}}}={\sqrt[{4}]{\frac {2143}{22}}}=3{,}14159\ 2652^{+}}" /></span></dd></dl></dd></dl> <ul><li>yhdeksän oikeaa desimaalia:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {63}{25}}\cdot {\frac {17+15{\sqrt {5}}}{7+15{\sqrt {5}}}}=3{,}14159\ 26538^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>63</mn> <mn>25</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>17</mn> <mo>+</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mrow> <mn>7</mn> <mo>+</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>3,141</mn> <mn>59</mn> <mtext> </mtext> <msup> <mn>26538</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {63}{25}}\cdot {\frac {17+15{\sqrt {5}}}{7+15{\sqrt {5}}}}=3{,}14159\ 26538^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f4998574df51fa036e3b0a3aa8d4f6b9727b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.889ex; height:6.843ex;" alt="{\displaystyle {\frac {63}{25}}\cdot {\frac {17+15{\sqrt {5}}}{7+15{\sqrt {5}}}}=3{,}14159\ 26538^{+}}" /></span></dd></dl></dd></dl> <ul><li>yhdeksän oikeaa desimaalia:</li></ul> <dl><dd><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 {\sqrt[{193}]{\frac {10^{100}}{11222{,}11122}}}=3{,}14159\ 26536^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>100</mn> </mrow> </msup> <mrow> <mn>11222,111</mn> <mn>22</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>193</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mn>3,141</mn> <mn>59</mn> <mtext> </mtext> <msup> <mn>26536</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{193}]{\frac {10^{100}}{11222{,}11122}}}=3{,}14159\ 26536^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5f9747a67502acbebcf60e0b70c01e75c878fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.489ex; height:7.676ex;" alt="{\displaystyle {\sqrt[{193}]{\frac {10^{100}}{11222{,}11122}}}=3{,}14159\ 26536^{+}}" /></span></dd></dl></dd></dl> <ul><li>17 oikeaa desimaalia:</li></ul> <dl><dd><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 {80{\sqrt {15}}(5^{4}+53{\sqrt {89}})^{\frac {3}{2}}}{3308(5^{4}+53{\sqrt {89}})-3{\sqrt {89}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>80</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>53</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>89</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mn>3308</mn> <mo stretchy="false">(</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>53</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>89</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>89</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {80{\sqrt {15}}(5^{4}+53{\sqrt {89}})^{\frac {3}{2}}}{3308(5^{4}+53{\sqrt {89}})-3{\sqrt {89}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508fe04d39ea79dea7625969e1403cc17b410bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.202ex; height:8.009ex;" alt="{\displaystyle {\frac {80{\sqrt {15}}(5^{4}+53{\sqrt {89}})^{\frac {3}{2}}}{3308(5^{4}+53{\sqrt {89}})-3{\sqrt {89}}}}}" /></span></dd></dl></dd></dl> <ul><li>29 oikeaa desimaalia:</li></ul> <dl><dd><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 {\ln(640320^{3}+744)}{\sqrt {163}}}=3{,}14159\ 26535\ 89793\ 23846\ 26433\ 83279^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>640320</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>744</mn> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>163</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>3,141</mn> <mn>59</mn> <mtext> </mtext> <mn>26535</mn> <mtext> </mtext> <mn>89793</mn> <mtext> </mtext> <mn>23846</mn> <mtext> </mtext> <mn>26433</mn> <mtext> </mtext> <msup> <mn>83279</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ln(640320^{3}+744)}{\sqrt {163}}}=3{,}14159\ 26535\ 89793\ 23846\ 26433\ 83279^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da841fb8bd9cf071d5d0252e3e181401a6d2e48a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.137ex; height:6.843ex;" alt="{\displaystyle {\frac {\ln(640320^{3}+744)}{\sqrt {163}}}=3{,}14159\ 26535\ 89793\ 23846\ 26433\ 83279^{+}}" /></span></dd></dl></dd></dl> <ul><li>51 oikeaa desimaalia:</li></ul> <dl><dd><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 {\ln(5280^{3}(236674+30303{\sqrt {61}})^{3}+744)}{\sqrt {427}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>5280</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>236674</mn> <mo>+</mo> <mn>30303</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>61</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>744</mn> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>427</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ln(5280^{3}(236674+30303{\sqrt {61}})^{3}+744)}{\sqrt {427}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985ff8321678c2db7d8c3332590e1523ef00473e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.368ex; height:7.009ex;" alt="{\displaystyle {\frac {\ln(5280^{3}(236674+30303{\sqrt {61}})^{3}+744)}{\sqrt {427}}}}" /></span></dd></dl></dd></dl> <ul><li>160 oikeaa desimaalia:</li></ul> <dl><dd><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 {\ln {\big (}(2u)^{6}+24{\big )}}{\sqrt {3502}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msqrt> <mn>3502</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ln {\big (}(2u)^{6}+24{\big )}}{\sqrt {3502}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e22c62f18020f508a557a0527f97654fd6838ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.813ex; height:7.009ex;" alt="{\displaystyle {\frac {\ln {\big (}(2u)^{6}+24{\big )}}{\sqrt {3502}}}}" /></span></dd></dl></dd></dl> <dl><dd>missä</dd></dl> <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 u=(a+{\sqrt {a^{2}-1}})^{2}(b+{\sqrt {b^{2}-1}})^{2}(c+{\sqrt {c^{2}-1}})(d+{\sqrt {d^{2}-1}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=(a+{\sqrt {a^{2}-1}})^{2}(b+{\sqrt {b^{2}-1}})^{2}(c+{\sqrt {c^{2}-1}})(d+{\sqrt {d^{2}-1}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0192c6320bdfe6df70630d76144a7ba841e938c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.561ex; height:3.509ex;" alt="{\displaystyle u=(a+{\sqrt {a^{2}-1}})^{2}(b+{\sqrt {b^{2}-1}})^{2}(c+{\sqrt {c^{2}-1}})(d+{\sqrt {d^{2}-1}})}" /></span></dd></dl> <dl><dd>ja</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&={\tfrac {1}{2}}(23+4{\sqrt {34}})\\b&={\tfrac {1}{2}}(19{\sqrt {2}}+7{\sqrt {17}})\\c&=(429+304{\sqrt {2}})\\d&={\tfrac {1}{2}}(627+442{\sqrt {2}})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>23</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>19</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>429</mn> <mo>+</mo> <mn>304</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>627</mn> <mo>+</mo> <mn>442</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&={\tfrac {1}{2}}(23+4{\sqrt {34}})\\b&={\tfrac {1}{2}}(19{\sqrt {2}}+7{\sqrt {17}})\\c&=(429+304{\sqrt {2}})\\d&={\tfrac {1}{2}}(627+442{\sqrt {2}})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f316c23aa50198736b2355a0183a9ba200a2086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:22.234ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}a&={\tfrac {1}{2}}(23+4{\sqrt {34}})\\b&={\tfrac {1}{2}}(19{\sqrt {2}}+7{\sqrt {17}})\\c&=(429+304{\sqrt {2}})\\d&={\tfrac {1}{2}}(627+442{\sqrt {2}})\end{aligned}}}" /></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Piin_desimaalien_laskeminen">Piin desimaalien laskeminen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=5" title="Muokkaa osiota Piin desimaalien laskeminen" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=5" title="Muokkaa osion lähdekoodia: Piin desimaalien laskeminen"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Englantilainen amatöörimatemaatikko <a href="/w/index.php?title=William_Shanks&action=edit&redlink=1" class="new" title="William Shanks (sivua ei ole)">William Shanks</a> (1812 – 1882) laski 20 vuoden ajan piin desimaaleja käsin ja ratkaisi luvun 707 desimaalin tarkkuudella. Vuonna 1945 kuitenkin huomattiin, että laskun 528. desimaali oli laskettu virheellisesti.<sup id="cite_ref-T_7-0" class="reference"><a href="#cite_note-T-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Nykyään piin desimaaleja lasketaan tietokoneilla. </p><p>1900-luvulla pii tunnettiin jo yli miljardin desimaalin tarkkuudella, ja elokuussa 2021 siitä tiedettiin ensimmäiset 62,8 biljoonaa desimaalia.<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 id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Vuoden 2024 keväällä piille laskettiin 105 <a href="/wiki/Biljoona" title="Biljoona">biljoonaa</a> eli 1 05 000 000 000 000 desimaalia.<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><p>1990-luvulla kehitettiin tapoja laskea piin <a href="/wiki/Heksadesimaali" class="mw-redirect" title="Heksadesimaali">heksadesimaaliesityksen</a> numeroita, mistä tahansa kohdasta ilman, että aiempia numeroita tarvitsee tietää. </p> <div class="mw-heading mw-heading2"><h2 id="Muita_esimerkkejä"><span id="Muita_esimerkkej.C3.A4"></span>Muita esimerkkejä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=6" title="Muokkaa osiota Muita esimerkkejä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=6" title="Muokkaa osion lähdekoodia: Muita esimerkkejä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:Ympyr%C3%A4_neli%C3%B6ruudukolla.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/fi/3/39/Ympyr%C3%A4_neli%C3%B6ruudukolla.jpg" decoding="async" width="239" height="239" class="mw-file-element" data-file-width="239" data-file-height="239" /></a><figcaption>Kolikkoa kuvaava ympyrä neliöruudukolla.</figcaption></figure> <p>Kun kolikko heitetään satunnaisesti neliöruudukolle, jossa kunkin neliön sivun pituus on sama kuin kolikon halkaisija (eli kaksi kertaa säde r), niin todennäköisyys, että kolikko peittää neliöiden risteyskohdan kuvassa esitetyllä tavalla, on π/4. Toistamalla koe lukuisia kertoja saadaan piille kokeellinen likiarvo.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Avoimia_kysymyksiä"><span id="Avoimia_kysymyksi.C3.A4"></span>Avoimia kysymyksiä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=7" title="Muokkaa osiota Avoimia kysymyksiä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=7" title="Muokkaa osion lähdekoodia: Avoimia kysymyksiä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vaikka piistä tiedetään paljon, on vielä useita avoimia kysymyksiä sen desimaaleihin liittyen: </p> <ul><li>Onko desimaaleissa toistuvia kuvioita, vai onko ketju hahmoton?<sup id="cite_ref-T_7-1" class="reference"><a href="#cite_note-T-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>Toistuvatko jotkin numerot tai luvut piissä useammin kuin toiset?<sup id="cite_ref-T_7-2" class="reference"><a href="#cite_note-T-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=8" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=8" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Luettelo_piin_laskukaavoista" title="Luettelo piin laskukaavoista">Luettelo piin laskukaavoista</a></li> <li><a href="/wiki/Feynmanin_piste" title="Feynmanin piste">Feynmanin piste</a></li> <li><a href="/wiki/Tau_(vakio)" title="Tau (vakio)">Tau (vakio)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=9" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=9" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="kirjaviite" title="Kirjaviite">Pappas, Theoni: <i>Lisää matematiikan iloja</i>.  (Alkuteos: More Joy of Mathematics. Exploring Mathematics All Around You)  Suomentanut Juha Pietiläinen. Helsinki:  Terra Cognita, 1991.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/952-5202-46-1" title="Toiminnot:Kirjalähteet/952-5202-46-1">ISBN 952-5202-46-1</a> </span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Viitteet">Viitteet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=10" title="Muokkaa osiota Viitteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=10" title="Muokkaa osion lähdekoodia: Viitteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23091323">.mw-parser-output div.viitteet-malline ol.references{list-style-type:inherit}.mw-parser-output .viitteet-sarakkeet li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output ol.references>li:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output sup.reference:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output span[rel="mw:referencedBy"]{counter-reset:mw-ref-linkback 0}.mw-parser-output span[rel="mw:referencedBy"]>a::before{font-style:italic;content:counter(mw-ref-linkback,lower-alpha)}body.action-info .mw-parser-output :target{background-color:var(--background-color-interactive,#DEF)}</style><div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-archimedes-1"><span class="mw-cite-backlink"><a href="#cite_ref-archimedes_1-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ArchimedesConstant.html">Archimedes' Constant</a>, Wolfram Research, viitattu 10.2.2021 <span style="font-size: 0.95em; position: relative;">(englanniksi)</span></span> </li> <li id="cite_note-ludolph-2"><span class="mw-cite-backlink"><a href="#cite_ref-ludolph_2-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LudolphsConstant.html">Ludolph's Constant</a>, Wolfram Research, viitattu 10.2.2021 <span style="font-size: 0.95em; position: relative;">(englanniksi)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://raamattu.uskonkirjat.net/servlet/biblesite.Bible?ref=1.+Kun.+7%3A23&mod1=FinRaam&mod2=FinPR&mod3=YLT&ctx=0">1. Kun. 7:23</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Iso tietosanakirja, 9. osa (Mustonen-Pielisjärvi), art. Pi, Otava 1935</span> </li> <li id="cite_note-JM-5"><span class="mw-cite-backlink">↑ <a href="#cite_ref-JM_5-0"><sup><i>a</i></sup></a> <a href="#cite_ref-JM_5-1"><sup><i>b</i></sup></a> <a href="#cite_ref-JM_5-2"><sup><i>c</i></sup></a></span> <span class="reference-text"><span class="kirjaviite" title="Kirjaviite">Jukka Männistö: <i>Matematiikan helmiä lukiolaisille</i>.  (Luvun pii määrittämiskeinoja, s. 17–23)  Tampereen yliopisto. Tampereen normaalikoulun julkaisuja. Sarja A1: Tutkielmia ja monisteita 1, 1993.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-44-3422-6" title="Toiminnot:Kirjalähteet/951-44-3422-6">ISBN 951-44-3422-6</a> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110706215615/http://www.mschneider.cc/papers/pi.pdf">A nested radical approximation for π</a></span> </li> <li id="cite_note-T-7"><span class="mw-cite-backlink">↑ <a href="#cite_ref-T_7-0"><sup><i>a</i></sup></a> <a href="#cite_ref-T_7-1"><sup><i>b</i></sup></a> <a href="#cite_ref-T_7-2"><sup><i>c</i></sup></a></span> <span class="reference-text">Theoni Pappas s. 45</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="verkkoviite" title="Verkkoviite"><a rel="nofollow" class="external text" href="https://fin.afterdawn.com/uutiset/artikkeli.cfm/2021/08/18/pii-likiarvo-ennatys">3,14.. Piille laskettiin nyt reipas määrä desimaaleja: 62 biljoonaa numeroa</a> <i>AfterDawn</i>. Viitattu 18.8.2021.</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 class="verkkoviite" title="Verkkoviite"><a rel="nofollow" class="external text" href="https://www.fhgr.ch/fachgebiete/angewandte-zukunftstechnologien/davis-zentrum/pi-challenge/">Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden</a> <i>www.fhgr.ch</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210818045706/https://www.fhgr.ch/fachgebiete/angewandte-zukunftstechnologien/davis-zentrum/pi-challenge/">Arkistoitu</a> 18.8.2021. Viitattu 18.8.2021.</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="verkkoviite" title="Verkkoviite">Tojkander, Maria: <a rel="nofollow" class="external text" href="https://tekniikanmaailma.fi/lehti/8b-2024/kuinka-monta-piin-desimaalia-on-pystytty-laskemaan/">Kuinka monta piin desimaalia on pystytty laskemaan?</a> <i>tekniikanmaailma.fi</i>. Viitattu 23.4.2024.</span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Yrjö Juve, <i>Todennäköisyyslaskennan alkeita</i>, 5. painos, Kirjayhtymä, 1971, s. 24–25.</span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="Kirjallisuutta">Kirjallisuutta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=11" title="Muokkaa osiota Kirjallisuutta" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=11" title="Muokkaa osion lähdekoodia: Kirjallisuutta"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="kirjaviite" title="Kirjaviite">Beckmann, Petr: <i>π: Erään luvun tarina</i>.  ((A history of π, 1971.) Suomentanut Hannele Salminen)  Helsinki:  Terra Cognita, 2000.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/952-5202-28-3" title="Toiminnot:Kirjalähteet/952-5202-28-3">ISBN 952-5202-28-3</a> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Aiheesta_muualla">Aiheesta muualla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pii_(vakio)&veaction=edit&section=12" title="Muokkaa osiota Aiheesta muualla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pii_(vakio)&action=edit&section=12" title="Muokkaa osion lähdekoodia: Aiheesta muualla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r22431496">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r22718453">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/Tiedosto:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commonsissa</a> on kuvia tai muita tiedostoja aiheesta <b><a href="https://commons.wikimedia.org/wiki/Category:Pi" class="extiw" title="commons:Category:Pi">Pii (vakio)</a></b>.</div></div> </div> <ul><li><a rel="nofollow" class="external autonumber" href="http://www.archive.org/details/pi00050gut">[1]</a> Gutenberg-projektin teksti, jossa on ensimmäiset 10 miljoonaa desimaalia</li> <li><a rel="nofollow" class="external text" href="http://numbers.computation.free.fr/Constants/PiProgram/pifast.html">PiFastilla</a> voi itse laskea piin ja muiden vakioiden arvoja erittäin tarkasti</li> <li><a rel="nofollow" class="external text" href="http://pidifferent.pi.funpic.de/index-en.html">Pi-memory</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20080319051557/http://pidifferent.pi.funpic.de/index-en.html">Arkistoitu</a> – Internet Archive)</li> <li><a rel="nofollow" class="external autonumber" href="http://www.apfloat.org/apfloat_java/applet/pi.html">[2]</a> Pii-laskin, joka toimii selaimessa</li></ul> <p><i style="display:none; 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Pii (vakio) – Wikipedia