e (mathematical constant)

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href="#Exponential_growth_and_decay"> <div class="vector-toc-text"> 3.3 Exponential growth and decay </div> </a> <ul id="toc-Exponential_growth_and_decay-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standard_normal_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standard_normal_distribution"> <div class="vector-toc-text"> 3.4 Standard normal distribution </div> </a> <ul id="toc-Standard_normal_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derangements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derangements"> <div class="vector-toc-text"> 3.5 Derangements </div> </a> <ul id="toc-Derangements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimal_planning_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimal_planning_problems"> <div class="vector-toc-text"> 3.6 Optimal planning problems </div> </a> <ul id="toc-Optimal_planning_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymptotics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotics"> <div class="vector-toc-text"> 3.7 Asymptotics </div> </a> <ul id="toc-Asymptotics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculus"> <div class="vector-toc-text"> 4.1 Calculus </div> </a> <ul id="toc-Calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inequalities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inequalities"> <div class="vector-toc-text"> 4.2 Inequalities </div> </a> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponential-like_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponential-like_functions"> <div class="vector-toc-text"> 4.3 Exponential-like functions </div> </a> <ul id="toc-Exponential-like_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> 4.4 Number theory </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> 4.5 Complex numbers </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> 5 Representations </div> </a> <button aria-controls="toc-Representations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Representations subsection </button> <ul id="toc-Representations-sublist" class="vector-toc-list"> <li id="toc-Stochastic_representations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_representations"> <div class="vector-toc-text"> 5.1 Stochastic representations </div> </a> <ul id="toc-Stochastic_representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Known_digits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Known_digits"> <div class="vector-toc-text"> 5.2 Known digits </div> </a> <ul id="toc-Known_digits-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computing_the_digits" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computing_the_digits"> <div class="vector-toc-text"> 6 Computing the digits </div> </a> <ul id="toc-Computing_the_digits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_computer_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_computer_culture"> <div class="vector-toc-text"> 7 In computer culture </div> </a> <ul id="toc-In_computer_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 9 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">e (mathematical constant)</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 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Available in 88 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-88" aria-hidden="true" > 88 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/E_(wiskunde)" title="E (wiskunde) – Afrikaans" lang="af" hreflang="af" data-title="E (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A6%E1%8B%AD%E1%88%88%E1%88%AD_%E1%89%81%E1%8C%A5%E1%88%AD" title="ኦይለር ቁጥር – Amharic" lang="am" hreflang="am" data-title="ኦይለር ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="ه (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="ه (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_e" title="Numero e – Aragonese" lang="an" hreflang="an" data-title="Numero e" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_e" title="Númberu e – Asturian" lang="ast" hreflang="ast" data-title="Númberu e" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/E_(%C9%99d%C9%99d)" title="E (ədəd) – Azerbaijani" lang="az" hreflang="az" data-title="E (ədəd)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B0_%E0%A6%A7%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AC%E0%A6%95" title="অয়লারের ধ্রুবক – Bangla" lang="bn" hreflang="bn" data-title="অয়লারের ধ্রুবক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%95_(%D2%BB%D0%B0%D0%BD)" title="Е (һан) – Bashkir" lang="ba" hreflang="ba" data-title="Е (һан)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/E_(%D0%BB%D1%96%D0%BA)" title="E (лік) – Belarusian" lang="be" hreflang="be" data-title="E (лік)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D0%B5%D1%80%D0%BE%D0%B2%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Неперово число – Bulgarian" lang="bg" hreflang="bg" data-title="Неперово число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/E_(broj)" title="E (broj) – Bosnian" lang="bs" hreflang="bs" data-title="E (broj)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/E_(niver)" title="E (niver) – Breton" lang="br" hreflang="br" data-title="E (niver)" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_e" title="Nombre e – Catalan" lang="ca" hreflang="ca" data-title="Nombre e" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/E_(%D1%85%D0%B8%D1%81%D0%B5%D0%BF)" title="E (хисеп) – Chuvash" lang="cv" hreflang="cv" data-title="E (хисеп)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eulerovo_%C4%8D%C3%ADslo" title="Eulerovo číslo – Czech" lang="cs" hreflang="cs" data-title="Eulerovo číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/E_(tal)" title="E (tal) – Danish" lang="da" hreflang="da" data-title="E (tal)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eulersche_Zahl" title="Eulersche Zahl – German" lang="de" hreflang="de" data-title="Eulersche Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/E_(arv)" title="E (arv) – Estonian" lang="et" hreflang="et" data-title="E (arv)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/E_(%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="E (μαθηματική σταθερά) – Greek" lang="el" hreflang="el" data-title="E (μαθηματική σταθερά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e" title="Número e – Spanish" lang="es" hreflang="es" data-title="Número e" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/E_(matematiko)" title="E (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="E (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://eu.wikipedia.org/wiki/E_(zenbakia)" title="E (zenbakia) – Basque" lang="eu" hreflang="eu" data-title="E (zenbakia)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/E_(%D8%B9%D8%AF%D8%AF)" title="E (عدد) – Persian" lang="fa" hreflang="fa" data-title="E (عدد)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/E_(nombre)" title="E (nombre) – French" lang="fr" hreflang="fr" data-title="E (nombre)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_e" title="Número e – Galician" lang="gl" hreflang="gl" data-title="Número e" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/E_(%E6%95%B8%E5%AD%B8%E5%B8%B8%E6%95%B8)" title="E (數學常數) – Gan" lang="gan" hreflang="gan" data-title="E (數學常數)" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8%EC%9D%98_%EB%B0%91" title="자연로그의 밑 – Korean" lang="ko" hreflang="ko" data-title="자연로그의 밑" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/E_(%D5%A9%D5%AB%D5%BE)" title="E (թիվ) – Armenian" lang="hy" hreflang="hy" data-title="E (թիվ)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/E_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%A4%E0%A4%BE%E0%A4%82%E0%A4%95)" title="E (गणितीय नियतांक) – Hindi" lang="hi" hreflang="hi" data-title="E (गणितीय नियतांक)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/E_(matemati%C4%8Dka_konstanta)" title="E (matematička konstanta) – Croatian" lang="hr" hreflang="hr" data-title="E (matematička konstanta)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Nombro_di_Euler" title="Nombro di Euler – Ido" lang="io" hreflang="io" data-title="Nombro di Euler" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/E_(konstanta_matematika)" title="E (konstanta matematika) – Indonesian" lang="id" hreflang="id" data-title="E (konstanta matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/E_(constante_mathematic)" title="E (constante mathematic) – Interlingua" lang="ia" hreflang="ia" data-title="E (constante mathematic)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/E_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0ilegur_fasti)" title="E (stærðfræðilegur fasti) – Icelandic" lang="is" hreflang="is" data-title="E (stærðfræðilegur fasti)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/E_(costante_matematica)" title="E (costante matematica) – Italian" lang="it" hreflang="it" data-title="E (costante matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/E_(%D7%A7%D7%91%D7%95%D7%A2_%D7%9E%D7%AA%D7%9E%D7%98%D7%99)" title="E (קבוע מתמטי) – Hebrew" lang="he" hreflang="he" data-title="E (קבוע מתמטי)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%88_(%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4%E0%B2%A6_%E0%B2%B8%E0%B3%8D%E0%B2%A5%E0%B2%BF%E0%B2%B0%E0%B2%BE%E0%B2%82%E0%B2%95)" title="ಈ (ಗಣಿತದ ಸ್ಥಿರಾಂಕ) – Kannada" lang="kn" hreflang="kn" data-title="ಈ (ಗಣಿತದ ಸ್ಥಿರಾಂಕ)" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ka.wikipedia.org/wiki/E_(%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98)" title="E (რიცხვი) – Georgian" lang="ka" hreflang="ka" data-title="E (რიცხვი)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/E_%D1%81%D0%B0%D0%BD%D1%8B" title="E саны – Kazakh" lang="kk" hreflang="kk" data-title="E саны" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/E_(nonm)" title="E (nonm) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="E (nonm)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_e" title="Numerus e – Latin" lang="la" hreflang="la" data-title="Numerus e" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/E_(matem%C4%81tiska_konstante)" title="E (matemātiska konstante) – Latvian" lang="lv" hreflang="lv" data-title="E (matemātiska konstante)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/E_(skai%C4%8Dius)" title="E (skaičius) – Lithuanian" lang="lt" hreflang="lt" data-title="E (skaičius)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_e" title="Numero e – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero e" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://lmo.wikipedia.org/wiki/E_(costant_matematega)" title="E (costant matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="E (costant matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-f%C3%A9le_sz%C3%A1m" title="Euler-féle szám – Hungarian" lang="hu" hreflang="hu" data-title="Euler-féle szám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mk.wikipedia.org/wiki/%D0%95_(%D0%B1%D1%80%D0%BE%D1%98)" title="Е (број) – Macedonian" lang="mk" hreflang="mk" data-title="Е (број)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/E_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="E (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="E (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%88_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80_%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="ई (गणिती स्थिरांक) – Marathi" lang="mr" hreflang="mr" data-title="ई (गणिती स्थिरांक)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/E_(pemalar)" title="E (pemalar) – Malay" lang="ms" hreflang="ms" data-title="E (pemalar)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B8%D0%B9%D0%BD_%D1%82%D0%BE%D0%BE" title="Эйлерийн тоо – Mongolian" lang="mn" hreflang="mn" data-title="Эйлерийн тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/E_(wiskunde)" title="E (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="E (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%27e%27_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%85%E0%A4%9A%E0%A4%B0)" title="'e' (गणितीय अचर) – Nepali" lang="ne" hreflang="ne" data-title="'e' (गणितीय अचर)" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8D%E3%82%A4%E3%83%94%E3%82%A2%E6%95%B0" title="ネイピア数 – Japanese" lang="ja" hreflang="ja" data-title="ネイピア数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Euler_sin_taal" title="Euler sin taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Euler sin taal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eulers_tall" title="Eulers tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Eulers tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/E_i_matematikk" title="E i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="E i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/E_(nombre)" title="E (nombre) – Occitan" lang="oc" hreflang="oc" data-title="E (nombre)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Nomba_e" title="Nomba e – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nomba e" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podstawa_logarytmu_naturalnego" title="Podstawa logarytmu naturalnego – Polish" lang="pl" hreflang="pl" data-title="Podstawa logarytmu naturalnego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/E_(constante_matem%C3%A1tica)" title="E (constante matemática) – Portuguese" lang="pt" hreflang="pt" data-title="E (constante matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/E_(constant%C4%83_matematic%C4%83)" title="E (constantă matematică) – Romanian" lang="ro" hreflang="ro" data-title="E (constantă matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/E_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="E (число) – Russian" lang="ru" hreflang="ru" data-title="E (число)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/E_(mathematical_constant)" title="E (mathematical constant) – Scots" lang="sco" hreflang="sco" data-title="E (mathematical constant)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numri_e" title="Numri e – Albanian" lang="sq" hreflang="sq" data-title="Numri e" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Basi_naturali_d%C3%AE_logaritmi" title="Basi naturali dî logaritmi – Sicilian" lang="scn" hreflang="scn" data-title="Basi naturali dî logaritmi" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/E_(%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%AD%E0%B6%BA)" title="E (ගණිත නියතය) – Sinhala" lang="si" hreflang="si" data-title="E (ගණිත නියතය)" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/E_(mathematical_constant)" title="E (mathematical constant) – Simple English" lang="en-simple" hreflang="en-simple" data-title="E (mathematical constant)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Eulerovo_%C4%8D%C3%ADslo" title="Eulerovo číslo – Slovak" lang="sk" hreflang="sk" data-title="Eulerovo číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/E_(matemati%C4%8Dna_konstanta)" title="E (matematična konstanta) – Slovenian" lang="sl" hreflang="sl" data-title="E (matematična konstanta)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_e" title="ژمارەی e – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی e" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/E_(%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0)" title="E (константа) – Serbian" lang="sr" hreflang="sr" data-title="E (константа)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Broj_e" title="Broj e – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Broj e" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neperin_luku" title="Neperin luku – Finnish" lang="fi" hreflang="fi" data-title="Neperin luku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/E_(tal)" title="E (tal) – Swedish" lang="sv" hreflang="sv" data-title="E (tal)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/E_(%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="E (கணித மாறிலி) – Tamil" lang="ta" hreflang="ta" data-title="E (கணித மாறிலி)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/E_(am%E1%B8%8Dan_n_tusnakt)" title="E (amḍan n tusnakt) – Tachelhit" lang="shi" hreflang="shi" data-title="E (amḍan n tusnakt)" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/E_(%D1%81%D0%B0%D0%BD)" title="E (сан) – Tatar" lang="tt" hreflang="tt" data-title="E (сан)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/E_(%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="E (ค่าคงตัว) – Thai" lang="th" hreflang="th" data-title="E (ค่าคงตัว)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%9D%D0%B5%D0%BF%D0%B5%D1%80" title="Адади Непер – Tajik" lang="tg" hreflang="tg" data-title="Адади Непер" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/E_say%C4%B1s%C4%B1" title="E sayısı – Turkish" lang="tr" hreflang="tr" data-title="E sayısı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/E_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="E (число) – Ukrainian" lang="uk" hreflang="uk" data-title="E (число)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/E_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AF%D8%A7%D8%A6%D9%85)" title="E (ریاضیاتی دائم) – Urdu" lang="ur" hreflang="ur" data-title="E (ریاضیاتی دائم)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://vi.wikipedia.org/wiki/E_(s%E1%BB%91)" title="E (số) – Vietnamese" lang="vi" hreflang="vi" data-title="E (số)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%AD%90%E6%8B%89%E6%95%B8" title="歐拉數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="歐拉數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/E%EF%BC%88%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0%EF%BC%89" 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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">2.71828..., base of natural logarithms</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses of "e", see <a href="/wiki/E_(disambiguation)" class="mw-disambig" title="E (disambiguation)">e (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Euler's number" redirects here. For other Euler's numbers, see <a href="/wiki/List_of_things_named_after_Leonhard_Euler#Numbers" class="mw-redirect" title="List of things named after Leonhard Euler">List of things named after Leonhard Euler § Numbers</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Constant value used in mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Euler's number</th></tr><tr><td colspan="2" class="infobox-subheader">e 2.71828...<a href="#cite_note-OEIS_decimal_expansion-1">[1]</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">History</th></tr><tr><th scope="row" class="infobox-label">Discovered</th><td class="infobox-data">1685</td></tr><tr><th scope="row" class="infobox-label">By</th><td class="infobox-data"><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></td></tr><tr><th scope="row" class="infobox-label">First mention</th><td class="infobox-data">Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685</td></tr><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"> <ul><li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li></ul> </div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hyperbola_E.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Hyperbola_E.svg/237px-Hyperbola_E.svg.png" decoding="async" width="237" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Hyperbola_E.svg/356px-Hyperbola_E.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Hyperbola_E.svg/474px-Hyperbola_E.svg.png 2x" data-file-width="609" data-file-height="652" /></a><figcaption>Graph of the equation y = 1/x. Here, e is the unique number larger than 1 that makes the shaded <a href="/wiki/Area_under_the_curve" class="mw-redirect" title="Area under the curve">area under the curve</a> equal to 1.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1238256010">.mw-parser-output .e-mathematical-constant-sidebar{width:20em}.mw-parser-output .e-mathematical-constant-sidebar .sidebar-title-with-pretitle{font-size:130%}.mw-parser-output .e-mathematical-constant-sidebar .sidebar-heading{border-top:#aaa 1px solid}</style><table class="sidebar nomobile nowraplinks hlist e-mathematical-constant-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:E_(mathematical_constant)" title="Category:E (mathematical constant)">a series of articles</a> on the</td></tr><tr><th class="sidebar-title-with-pretitle">mathematical constant <a class="mw-selflink selflink">e</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/180px-Euler%27s_formula.svg.png" decoding="async" width="180" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/270px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/360px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a></td></tr><tr><th class="sidebar-heading"> Properties</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Applications</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Compound_interest" title="Compound interest">compound interest</a></li> <li><a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a></li> <li><a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Half-life" title="Half-life">half-lives</a> <ul><li>exponential <a href="/wiki/Exponential_growth" title="Exponential growth">growth</a> and <a href="/wiki/Exponential_decay" title="Exponential decay">decay</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Defining e</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">proof that e is irrational</a></li> <li><a href="/wiki/List_of_representations_of_e" title="List of representations of e">representations of e</a></li> <li><a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> People</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> </li></ul></td> </tr><tr><th class="sidebar-heading"> Related topics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:E_(mathematical_constant)" title="Template:E (mathematical constant)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:E_(mathematical_constant)" title="Template talk:E (mathematical constant)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:E_(mathematical_constant)" title="Special:EditPage/Template:E (mathematical constant)"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> The number e is a <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a> approximately equal to 2.71828 that is the <a href="/wiki/Base_of_a_logarithm" class="mw-redirect" title="Base of a logarithm">base</a> of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> and <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>. It is sometimes called Euler's number, after the Swiss mathematician <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, though this can invite confusion with <a href="/wiki/Euler_numbers" title="Euler numbers">Euler numbers</a>, or with <a href="/wiki/Euler%27s_constant" title="Euler's constant">Euler's constant</a>, a different constant typically denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }">. Alternatively, e can be called Napier's constant after <a href="/wiki/John_Napier" title="John Napier">John Napier</a>.<a href="#cite_note-Miller-2">[2]</a><a href="#cite_note-Weisstein-3">[3]</a> The Swiss mathematician <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> discovered the constant while studying compound interest.<a href="#cite_note-Pickover-4">[4]</a><a href="#cite_note-OConnor-5">[5]</a> The number e is of great importance in mathematics,<a href="#cite_note-6">[6]</a> alongside 0, 1, <a href="/wiki/Pi" title="Pi">π</a>, and <a href="/wiki/Imaginary_unit" title="Imaginary unit">i</a>. All five appear in one formulation of <a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi }+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\pi }+1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7464809a40f9e486de3a454745f572fbf8bb256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.089ex; height:2.843ex;" alt="{\displaystyle e^{i\pi }+1=0}"> and play important and recurring roles across mathematics.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> Like the constant π, e is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>, meaning that it cannot be represented as a ratio of integers, and moreover it is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>, meaning that it is not a root of any non-zero <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with rational coefficients.<a href="#cite_note-Weisstein-3">[3]</a> To 30 decimal places, the value of e is:<a href="#cite_note-OEIS_decimal_expansion-1">[1]</a> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">2.718281828459045235360287471352 </div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=1" title="Edit section: Definitions">edit</a>]</div> The number e is the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abb8f4094bafd4757b3e350336fbc66a4b67675" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.18ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}"> an expression that arises in the computation of <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>. It is the sum of the infinite <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <munderover> <mo movablelimits="false">∑</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> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <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> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9a1f86072b07e1f69d5e21571c207d52680d8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.702ex; height:6.843ex;" alt="{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}"> It is the unique positive number a such that the graph of the function y = ax has a <a href="/wiki/Slope" title="Slope">slope</a> of 1 at x = 0. One has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\exp(1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\exp(1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5788ee8a9b7d2f155f4c635142cf74879daea13f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.353ex; height:2.843ex;" alt="{\displaystyle e=\exp(1),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1185b570f67b4221307626254f64f9e619e769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }"> is the (natural) <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, the unique function that equals its own <a href="/wiki/Derivative" title="Derivative">derivative</a> and satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(0)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(0)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a538ff408ca744879fb353be27dbfae4d511b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:2.843ex;" alt="{\displaystyle \exp(0)=1.}"> Since the exponential function is commonly denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto e^{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto e^{x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4abbbdd9b9faf016185bb1cd9d8ba9520ee0cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.847ex; height:2.676ex;" alt="{\displaystyle x\mapsto e^{x},}"> one has also <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=e^{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=e^{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5212cb82dee76925a5499d281b787c4ef9d262b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.967ex; height:2.676ex;" alt="{\displaystyle e=e^{1}.}"> The <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of base b can be defined as the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto b^{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto b^{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20dff57d5ff150fcd5b1061cc34610a8e4572603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.761ex; height:2.343ex;" alt="{\displaystyle x\mapsto b^{x}.}"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=b^{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=b^{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0d8da70f119de4029d4581d2c59e33a914bbf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:3.009ex;" alt="{\displaystyle b=b^{1},}"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}b=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}b=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afd18263767c2274147a008174865fcf178ba84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.202ex; height:2.676ex;" alt="{\displaystyle \log _{b}b=1.}"> The equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=e^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=e^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2751ef5de2a81ebf127c01f3f57163e26867a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.32ex; height:2.676ex;" alt="{\displaystyle e=e^{1}}"> implies therefore that e is the base of the natural logarithm. The number e can also be characterized in terms of an <a href="/wiki/Integral" title="Integral">integral</a>:<a href="#cite_note-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{1}^{e}{\frac {dx}{x}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{1}^{e}{\frac {dx}{x}}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87097c80b75c01a2a898586ed94e83203f1804d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.139ex; height:5.843ex;" alt="{\displaystyle \int _{1}^{e}{\frac {dx}{x}}=1.}"> For other characterizations, see <a href="#Representations">§ Representations</a>. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=2" title="Edit section: History">edit</a>]</div> The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by <a href="/wiki/John_Napier" title="John Napier">John Napier</a>. However, this did not contain the constant itself, but simply a list of <a href="/wiki/Natural_logarithm" title="Natural logarithm">logarithms to the base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></a>. It is assumed that the table was written by <a href="/wiki/William_Oughtred" title="William Oughtred">William Oughtred</a>. In 1661, <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of e, but he did not recognize e itself as a quantity of interest.<a href="#cite_note-OConnor-5">[5]</a><a href="#cite_note-10">[10]</a> The constant itself was introduced by <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> in 1683, for solving the problem of <a href="/wiki/Continuous_compounding" class="mw-redirect" title="Continuous compounding">continuous compounding</a> of interest.<a href="#cite_note-Bernoulli,_1690-11">[11]</a><a href="#cite_note-12">[12]</a> In his solution, the constant e occurs as the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abb8f4094bafd4757b3e350336fbc66a4b67675" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.18ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},}"> where n represents the number of intervals in a year on which the compound interest is evaluated (for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38fb7dfd38b5db65d78bb6f2c1bf9e6e6cd07c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=12}"> for monthly compounding). The first symbol used for this constant was the letter b by <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> in letters to <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> in 1690 and 1691.<a href="#cite_note-13">[13]</a> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,<a href="#cite_note-Meditatio-14">[14]</a> and in a letter to <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a> on 25 November 1731.<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> The first appearance of e in a printed publication was in Euler's <a href="/wiki/Mechanica" title="Mechanica">Mechanica</a> (1736).<a href="#cite_note-17">[17]</a> It is unknown why Euler chose the letter e.<a href="#cite_note-18">[18]</a> Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard.<a href="#cite_note-Miller-2">[2]</a> Euler proved that e is the sum of the <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fea921e7c8338dff877cc756a5d88ccbbb3cb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.086ex; height:6.843ex;" alt="{\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots ,}"> where n! is the <a href="/wiki/Factorial" title="Factorial">factorial</a> of n.<a href="#cite_note-OConnor-5">[5]</a> The equivalence of the two characterizations using the limit and the infinite series can be proved via the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>.<a href="#cite_note-19">[19]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=3" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Compound_interest">Compound interest</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=4" title="Edit section: Compound interest">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Compound_Interest_with_Varying_Frequencies.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/220px-Compound_Interest_with_Varying_Frequencies.svg.png" decoding="async" width="220" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/330px-Compound_Interest_with_Varying_Frequencies.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/440px-Compound_Interest_with_Varying_Frequencies.svg.png 2x" data-file-width="900" data-file-height="560" /></a><figcaption>The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies. The limiting curve on top is the graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=1000e^{0.2t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>1000</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.2</mn> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=1000e^{0.2t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/041055b6a4e391e953e4b619ee4d4c53bb090a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.915ex; height:3.009ex;" alt="{\displaystyle y=1000e^{0.2t}}">, where y is in dollars, t in years, and 0.2 = 20%.</figcaption></figure> Jacob Bernoulli discovered this constant in 1683, while studying a question about <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>:<a href="#cite_note-OConnor-5">[5]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?</blockquote> If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035.... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00 × (1 + 1/n)n.<a href="#cite_note-Gonick-20">[20]</a><a href="#cite_note-:0-21">[21]</a> Bernoulli noticed that this sequence approaches a limit (the <a href="/wiki/Force_of_interest" class="mw-redirect" title="Force of interest">force of interest</a>) with larger n and, thus, smaller compounding intervals.<a href="#cite_note-OConnor-5">[5]</a> Compounding weekly (n = 52) yields $2.692596..., while compounding daily (n = 365) yields $2.714567... (approximately two cents more). The limit as n grows large is the number that came to be known as e. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. Here, R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.<a href="#cite_note-Gonick-20">[20]</a><a href="#cite_note-:0-21">[21]</a> <div class="mw-heading mw-heading3"><h3 id="Bernoulli_trials">Bernoulli trials</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=5" title="Edit section: Bernoulli trials">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bernoulli_trial_sequence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Bernoulli_trial_sequence.svg/300px-Bernoulli_trial_sequence.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Bernoulli_trial_sequence.svg/450px-Bernoulli_trial_sequence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Bernoulli_trial_sequence.svg/600px-Bernoulli_trial_sequence.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 − P  vs n ; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.</figcaption></figure> The number e itself also has applications in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. As n increases, the probability that gambler will lose all n bets approaches 1/e. For n = 20, this is already approximately 1/2.789509.... This is an example of a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a> process. Each time the gambler plays the slots, there is a one in n chance of winning. Playing n times is modeled by the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, which is closely related to the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> and <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. The probability of winning k times out of n trials is:<a href="#cite_note-22">[22]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[k~\mathrm {wins~of} ~n]={\binom {n}{k}}\left({\frac {1}{n}}\right)^{k}\left(1-{\frac {1}{n}}\right)^{n-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mtext> </mtext> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[k~\mathrm {wins~of} ~n]={\binom {n}{k}}\left({\frac {1}{n}}\right)^{k}\left(1-{\frac {1}{n}}\right)^{n-k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11467bf6b08712fd2081a929d154c3e1c17f07dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.008ex; height:6.676ex;" alt="{\displaystyle \Pr[k~\mathrm {wins~of} ~n]={\binom {n}{k}}\left({\frac {1}{n}}\right)^{k}\left(1-{\frac {1}{n}}\right)^{n-k}.}"></dd></dl> In particular, the probability of winning zero times (k = 0) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[0~\mathrm {wins~of} ~n]=\left(1-{\frac {1}{n}}\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mtext> </mtext> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[0~\mathrm {wins~of} ~n]=\left(1-{\frac {1}{n}}\right)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534de6d23bfaf39e681e5d8cf2724fa19fc687e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.267ex; height:6.176ex;" alt="{\displaystyle \Pr[0~\mathrm {wins~of} ~n]=\left(1-{\frac {1}{n}}\right)^{n}.}"></dd></dl> The limit of the above expression, as n tends to infinity, is precisely 1/e. <div class="mw-heading mw-heading3"><h3 id="Exponential_growth_and_decay">Exponential growth and decay</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=6" title="Edit section: Exponential growth and decay">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Exponential_growth" title="Exponential growth">Exponential growth</a> and <a href="/wiki/Exponential_decay" title="Exponential decay">Exponential decay</a></div> <a href="/wiki/Exponential_growth" title="Exponential growth">Exponential growth</a> is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous <a href="/wiki/Rate_(mathematics)#Of_change" title="Rate (mathematics)">rate of change</a> (that is, the <a href="/wiki/Derivative" title="Derivative">derivative</a>) of a quantity with respect to time is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the quantity itself.<a href="#cite_note-:0-21">[21]</a> Described as a function, a quantity undergoing exponential growth is an <a href="/wiki/Exponentiation#Power_functions" title="Exponentiation">exponential function</a> of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as <a href="/wiki/Quadratic_growth" title="Quadratic growth">quadratic growth</a>). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a> instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different <a href="/wiki/Exponentiation" title="Exponentiation">base</a>, for which the number e is a common and convenient choice: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7372dd3a702bc6b5ccd275c4e7466e9c1634f204" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.296ex; height:3.343ex;" alt="{\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }.}"> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> denotes the initial value of the quantity x, k is the growth constant, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> is the time it takes the quantity to grow by a factor of e. <div class="mw-heading mw-heading3"><h3 id="Standard_normal_distribution">Standard normal distribution</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=7" title="Edit section: Standard normal distribution">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></div> The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution,<a href="#cite_note-openstax-23">[23]</a> given by the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4a5d73fb1296606c68401ae0ab4f9697b1b723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.406ex; height:6.176ex;" alt="{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}.}"> The constraint of unit standard deviation (and thus also unit variance) results in the <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠ in the exponent, and the constraint of unit total area under the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546b660b2f3cfb5f34be7b3ed8371d54f5c74227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}"> results in the factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1/{\sqrt {2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 1/{\sqrt {2\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc8e3e4df369ae48dea75e5d9ce018177064a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.755ex; height:3.176ex;" alt="{\displaystyle \textstyle 1/{\sqrt {2\pi }}}">. This function is symmetric around x = 0, where it attains its maximum value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1/{\sqrt {2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 1/{\sqrt {2\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc8e3e4df369ae48dea75e5d9ce018177064a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.755ex; height:3.176ex;" alt="{\displaystyle \textstyle 1/{\sqrt {2\pi }}}">, and has <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a> at x = ±1. <div class="mw-heading mw-heading3"><h3 id="Derangements">Derangements</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=8" title="Edit section: Derangements">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Derangement" title="Derangement">Derangement</a></div> Another application of e, also discovered in part by Jacob Bernoulli along with <a href="/wiki/Pierre_Remond_de_Montmort" title="Pierre Remond de Montmort">Pierre Remond de Montmort</a>, is in the problem of <a href="/wiki/Derangement" title="Derangement">derangements</a>, also known as the hat check problem:<a href="#cite_note-24">[24]</a> n guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into n boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. This probability, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d31a4c0444bd4002de6089482e2e7479d3784a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}\!}">, is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots +{\frac {(-1)^{n}}{n!}}=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <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>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots +{\frac {(-1)^{n}}{n!}}=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cd19cf6ff4b4104f532bd9c7d713d3e249b09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:52.626ex; height:7.176ex;" alt="{\displaystyle p_{n}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots +{\frac {(-1)^{n}}{n!}}=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.}"></dd></dl> As n tends to infinity, pn approaches 1/e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is n!/e, <a href="/wiki/Rounding" title="Rounding">rounded</a> to the nearest integer, for every positive n.<a href="#cite_note-25">[25]</a> <div class="mw-heading mw-heading3"><h3 id="Optimal_planning_problems">Optimal planning problems</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=9" title="Edit section: Optimal planning problems">edit</a>]</div> The maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{x}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{x}]{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b4a0a76158849854a302fc639dfc882ec16008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{x}]{x}}}"> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.512ex; height:1.676ex;" alt="{\displaystyle x=e}">. Equivalently, for any value of the base b > 1, it is the case that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}\log _{b}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}\log _{b}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef00edd9bb547a2e7f1587920bbcbe9e395391f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.676ex; height:3.176ex;" alt="{\displaystyle x^{-1}\log _{b}x}"> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.512ex; height:1.676ex;" alt="{\displaystyle x=e}"> (<a href="/wiki/Steiner%27s_calculus_problem" title="Steiner's calculus problem">Steiner's problem</a>, discussed <a href="#Exponential-like_functions">below</a>). This is useful in the problem of a stick of length L that is broken into n equal parts. The value of n that maximizes the product of the lengths is then either<a href="#cite_note-Finch-2003-p14-26">[26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\left\lfloor {\frac {L}{e}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>e</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\left\lfloor {\frac {L}{e}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558720ed36cc9ec4216745f558379040e7e3a155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.623ex; height:6.176ex;" alt="{\displaystyle n=\left\lfloor {\frac {L}{e}}\right\rfloor }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\lceil {\frac {L}{e}}\right\rceil .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>e</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lceil {\frac {L}{e}}\right\rceil .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1857ad26c882953299f989477879de56a1c84ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.163ex; height:6.176ex;" alt="{\displaystyle \left\lceil {\frac {L}{e}}\right\rceil .}"></dd></dl> The quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}\log _{b}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}\log _{b}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef00edd9bb547a2e7f1587920bbcbe9e395391f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.676ex; height:3.176ex;" alt="{\displaystyle x^{-1}\log _{b}x}"> is also a measure of <a href="/wiki/Shannon_information" class="mw-redirect" title="Shannon information">information</a> gleaned from an event occurring with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/x}"> (approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36.8\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36.8</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36.8\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d5bb204e1c9d1872ce0ef561dcc98807cda718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.343ex;" alt="{\displaystyle 36.8\%}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.512ex; height:1.676ex;" alt="{\displaystyle x=e}">), so that essentially the same optimal division appears in optimal planning problems like the <a href="/wiki/Secretary_problem" title="Secretary problem">secretary problem</a>. <div class="mw-heading mw-heading3"><h3 id="Asymptotics">Asymptotics</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=10" title="Edit section: Asymptotics">edit</a>]</div> The number e occurs naturally in connection with many problems involving <a href="/wiki/Asymptotics" class="mw-redirect" title="Asymptotics">asymptotics</a>. An example is <a href="/wiki/Stirling%27s_formula" class="mw-redirect" title="Stirling's formula">Stirling's formula</a> for the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotics</a> of the <a href="/wiki/Factorial_function" class="mw-redirect" title="Factorial function">factorial function</a>, in which both the numbers e and <a href="/wiki/Pi" title="Pi">π</a> appear:<a href="#cite_note-greg-27">[27]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3c28f23e205ed542a2b9bbeff5c56db3881877" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.837ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}"> As a consequence,<a href="#cite_note-greg-27">[27]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mroot> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0cd8eb0003d22f6e68f69c95cf434c43bebc58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.302ex; height:5.676ex;" alt="{\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}.}"> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=11" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Calculus">Calculus</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=12" title="Edit section: Calculus">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Characterizations_of_the_exponential_function" title="Characterizations of the exponential function">Characterizations of the exponential function</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Exp_derivative_at_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Exp_derivative_at_0.svg/220px-Exp_derivative_at_0.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Exp_derivative_at_0.svg/330px-Exp_derivative_at_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Exp_derivative_at_0.svg/440px-Exp_derivative_at_0.svg.png 2x" data-file-width="255" data-file-height="255" /></a><figcaption>The graphs of the functions x ↦ ax are shown for a = 2 (dotted), a = e (blue), and a = 4 (dashed). They all pass through the point (0,1), but the red line (which has slope 1) is tangent to only ex there.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ln%2Be.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Ln%2Be.svg/220px-Ln%2Be.svg.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Ln%2Be.svg/330px-Ln%2Be.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Ln%2Be.svg/440px-Ln%2Be.svg.png 2x" data-file-width="400" data-file-height="250" /></a><figcaption>The value of the natural log function for argument e, i.e. ln e, equals 1.</figcaption></figure> The principal motivation for introducing the number e, particularly in <a href="/wiki/Calculus" title="Calculus">calculus</a>, is to perform <a href="/wiki/Derivative_(mathematics)" class="mw-redirect" title="Derivative (mathematics)">differential</a> and <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a> with <a href="/wiki/Exponential_function" title="Exponential function">exponential functions</a> and <a href="/wiki/Logarithm" title="Logarithm">logarithms</a>.<a href="#cite_note-kline-28">[28]</a> A general exponential function y = ax has a derivative, given by a <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}a^{x}&=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=\lim _{h\to 0}{\frac {a^{x}a^{h}-a^{x}}{h}}\\&=a^{x}\cdot \left(\lim _{h\to 0}{\frac {a^{h}-1}{h}}\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi>h</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mi>h</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>h</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}a^{x}&=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=\lim _{h\to 0}{\frac {a^{x}a^{h}-a^{x}}{h}}\\&=a^{x}\cdot \left(\lim _{h\to 0}{\frac {a^{h}-1}{h}}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af4f2aadd037812a0daacfa5eea2a9e0ba091ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.327ex; margin-bottom: -0.177ex; width:41.927ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}a^{x}&=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=\lim _{h\to 0}{\frac {a^{x}a^{h}-a^{x}}{h}}\\&=a^{x}\cdot \left(\lim _{h\to 0}{\frac {a^{h}-1}{h}}\right).\end{aligned}}}"></dd></dl> The parenthesized limit on the right is independent of the variable x. Its value turns out to be the logarithm of a to base e. Thus, when the value of a is set to e, this limit is equal to 1, and so one arrives at the following simple identity: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0c27fbdebbf544403f5f1442939f21e7f77eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.639ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}"></dd></dl> Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler. Another motivation comes from considering the derivative of the base-a logarithm (i.e., loga x),<a href="#cite_note-kline-28">[28]</a> for x > 0: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\log _{a}x&=\lim _{h\to 0}{\frac {\log _{a}(x+h)-\log _{a}(x)}{h}}\\&=\lim _{h\to 0}{\frac {\log _{a}(1+h/x)}{x\cdot h/x}}\\&={\frac {1}{x}}\log _{a}\left(\lim _{u\to 0}(1+u)^{\frac {1}{u}}\right)\\&={\frac {1}{x}}\log _{a}e,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>⋅</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mi>e</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\log _{a}x&=\lim _{h\to 0}{\frac {\log _{a}(x+h)-\log _{a}(x)}{h}}\\&=\lim _{h\to 0}{\frac {\log _{a}(1+h/x)}{x\cdot h/x}}\\&={\frac {1}{x}}\log _{a}\left(\lim _{u\to 0}(1+u)^{\frac {1}{u}}\right)\\&={\frac {1}{x}}\log _{a}e,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b59e4b57e8dd05fe03460c15d45194bcf8b06fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.244ex; margin-bottom: -0.261ex; width:39.489ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\log _{a}x&=\lim _{h\to 0}{\frac {\log _{a}(x+h)-\log _{a}(x)}{h}}\\&=\lim _{h\to 0}{\frac {\log _{a}(1+h/x)}{x\cdot h/x}}\\&={\frac {1}{x}}\log _{a}\left(\lim _{u\to 0}(1+u)^{\frac {1}{u}}\right)\\&={\frac {1}{x}}\log _{a}e,\end{aligned}}}"></dd></dl> where the substitution u = h/x was made. The base-a logarithm of e is 1, if a equals e. So symbolically, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\log _{e}x={\frac {1}{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\log _{e}x={\frac {1}{x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a95626a9222ef0392a04642d314961425b37900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.367ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\log _{e}x={\frac {1}{x}}.}"></dd></dl> The logarithm with this special base is called the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>, and is usually denoted as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations. Thus, there are two ways of selecting such special numbers a. One way is to set the derivative of the exponential function ax equal to ax, and solve for a. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for a are actually the same: the number e. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Area_under_rectangular_hyperbola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Area_under_rectangular_hyperbola.svg/220px-Area_under_rectangular_hyperbola.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Area_under_rectangular_hyperbola.svg/330px-Area_under_rectangular_hyperbola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Area_under_rectangular_hyperbola.svg/440px-Area_under_rectangular_hyperbola.svg.png 2x" data-file-width="540" data-file-height="540" /></a><figcaption>The five colored regions are of equal area, and define units of <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> along the <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65451c8a210ae9635d8e1cfb592b8e3be0084a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.393ex; height:2.509ex;" alt="{\displaystyle xy=1.}"></figcaption></figure> The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0:<a href="#cite_note-strangherman-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2b1a7071d25d71cdd3b9e75ab6795938428d94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.128ex; height:6.843ex;" alt="{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}"> Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"> recovers the definition of e as the sum of an infinite series. The natural logarithm function can be defined as the integral from 1 to <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><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}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac33aabb19fba3d5d9b2e6008f61658ca2a3af3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.165ex; height:2.843ex;" alt="{\displaystyle 1/t}">, and the exponential function can then be defined as the inverse function of the natural logarithm. The number e is the value of the exponential function evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}">, or equivalently, the number whose natural logarithm is 1. It follows that e is the unique positive real number such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbefeb5e424ed3a0840aaedf1b792b2f94b9ee24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.199ex; height:5.843ex;" alt="{\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt=1.}"> Because ex is the unique function (<a href="/wiki/Up_to" title="Up to">up to</a> multiplication by a constant K) that is equal to its own <a href="/wiki/Derivative" title="Derivative">derivative</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}Ke^{x}=Ke^{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}Ke^{x}=Ke^{x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3f0b24ba62a1a1b2b58b4e3db87c1d86fdd898" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.771ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}Ke^{x}=Ke^{x},}"> it is therefore its own <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> as well:<a href="#cite_note-30">[30]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int Ke^{x}\,dx=Ke^{x}+C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int Ke^{x}\,dx=Ke^{x}+C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2728e875d935bf53c71ecd54a06b6c293410ed4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.509ex; height:5.676ex;" alt="{\displaystyle \int Ke^{x}\,dx=Ke^{x}+C.}"> Equivalently, the family of functions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)=Ke^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)=Ke^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4deac036ce10320199a532ae55193ff7e0873ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.715ex; height:2.843ex;" alt="{\displaystyle y(x)=Ke^{x}}"> where K is any real or complex number, is the full solution to the <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'=y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e049b1ccaa4b6e760b53b039bcd1ff95e21add2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.843ex;" alt="{\displaystyle y'=y.}"> <div class="mw-heading mw-heading3"><h3 id="Inequalities">Inequalities</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=13" title="Edit section: Inequalities">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Exponentials_vs_x%2B1.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Exponentials_vs_x%2B1.pdf/page1-220px-Exponentials_vs_x%2B1.pdf.jpg" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Exponentials_vs_x%2B1.pdf/page1-330px-Exponentials_vs_x%2B1.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Exponentials_vs_x%2B1.pdf/page1-440px-Exponentials_vs_x%2B1.pdf.jpg 2x" data-file-width="1702" data-file-height="1087" /></a><figcaption>Exponential functions y = 2x and y = 4x intersect the graph of y = x + 1, respectively, at x = 1 and x = -1/2. The number e is the unique base such that y = ex intersects only at x = 0. We may infer that e lies between 2 and 4.</figcaption></figure> The number e is the unique real number such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1+{\frac {1}{x}}\right)^{x}<e<\left(1+{\frac {1}{x}}\right)^{x+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo><</mo> <mi>e</mi> <mo><</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+{\frac {1}{x}}\right)^{x}<e<\left(1+{\frac {1}{x}}\right)^{x+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ceaacec4719cec85d5e528f3271317db096ef25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.906ex; height:6.509ex;" alt="{\displaystyle \left(1+{\frac {1}{x}}\right)^{x}<e<\left(1+{\frac {1}{x}}\right)^{x+1}}"> for all positive x.<a href="#cite_note-31">[31]</a> Also, we have the inequality <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}\geq x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>≥</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}\geq x+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b4efce8a73c717394d8c647e131c8cfe2067ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.687ex; height:2.509ex;" alt="{\displaystyle e^{x}\geq x+1}"> for all real x, with equality if and only if x = 0. Furthermore, e is the unique base of the exponential for which the inequality ax ≥ x + 1 holds for all x.<a href="#cite_note-32">[32]</a> This is a limiting case of <a href="/wiki/Bernoulli%27s_inequality" title="Bernoulli's inequality">Bernoulli's inequality</a>. <div class="mw-heading mw-heading3"><h3 id="Exponential-like_functions">Exponential-like functions</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=14" title="Edit section: Exponential-like functions">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Xth_root_of_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/250px-Xth_root_of_x.svg.png" decoding="async" width="250" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/375px-Xth_root_of_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/500px-Xth_root_of_x.svg.png 2x" data-file-width="512" data-file-height="465" /></a><figcaption>The <a href="/wiki/Global_maximum" class="mw-redirect" title="Global maximum">global maximum</a> of x√x occurs at x = e.</figcaption></figure> <a href="/wiki/Steiner%27s_calculus_problem" title="Steiner's calculus problem">Steiner's problem</a> asks to find the <a href="/wiki/Global_maximum" class="mw-redirect" title="Global maximum">global maximum</a> for the function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{\frac {1}{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{\frac {1}{x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e036e4a0d684bd9a12f61dff116387c5fc3b3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.324ex; height:3.843ex;" alt="{\displaystyle f(x)=x^{\frac {1}{x}}.}"> This maximum occurs precisely at x = e. (One can check that the derivative of ln f(x) is zero only for this value of x.) Similarly, x = 1/e is where the <a href="/wiki/Global_minimum" class="mw-redirect" title="Global minimum">global minimum</a> occurs for the function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6064c3a73cfc43c1b7e894a335566ae008d1ec68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.665ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{x}.}"> The infinite <a href="/wiki/Tetration" title="Tetration">tetration</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5280c74ea328a1153b7ae0eebd3571199eff2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.944ex; height:4.009ex;" alt="{\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {^{\infty }}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {^{\infty }}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/941e9b94ef87946de489befa76c6581c29c05df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.205ex; height:2.343ex;" alt="{\displaystyle {^{\infty }}x}"></dd></dl> converges if and only if x ∈ [(1/e)e, e1/e] ≈ [0.06599, 1.4447] ,<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a> shown by a theorem of <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.<a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=15" title="Edit section: Number theory">edit</a>]</div> The real number e is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> proved this by showing that its <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fraction</a> expansion does not terminate.<a href="#cite_note-38">[38]</a> (See also <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a>'s <a href="/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">proof that e is irrational</a>.) Furthermore, by the <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a>, e is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with <a href="/wiki/Liouville_number" title="Liouville number">Liouville number</a>); the proof was given by <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a> in 1873.<a href="#cite_note-39">[39]</a> The number e is one of only a few transcendental numbers for which the exact <a href="/wiki/Irrationality_measure#Irrationality_exponent" title="Irrationality measure">irrationality exponent</a> is known (given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (e)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (e)=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db1d1a2d408f5787506dd4e5350a47dc12800bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.555ex; height:2.843ex;" alt="{\displaystyle \mu (e)=2}">).<a href="#cite_note-40">[40]</a> An <a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">unsolved problem</a> thus far is the question of whether or not the numbers e and π are <a href="/wiki/Algebraic_independence" title="Algebraic independence">algebraically independent</a>. This would be resolved by <a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a> – a currently unproven generalization of the Lindemann–Weierstrass theorem.<a href="#cite_note-41">[41]</a><a href="#cite_note-42">[42]</a> It is conjectured that e is <a href="/wiki/Normal_number" title="Normal number">normal</a>, meaning that when e is expressed in any <a href="/wiki/Radix" title="Radix">base</a> the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).<a href="#cite_note-43">[43]</a> In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">period</a> is a number that can be expressed as an integral of an <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> over an algebraic <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>. The constant π is a period, but it is conjectured that e is not.<a href="#cite_note-44">[44]</a> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=16" title="Edit section: Complex numbers">edit</a>]</div> The <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> ex may be written as a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a><a href="#cite_note-45">[45]</a><a href="#cite_note-strangherman-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=1+{x \over 1!}+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=1+{x \over 1!}+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38978c994e4ef8842574d54e0890fd2a55a90fc7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.559ex; height:6.843ex;" alt="{\displaystyle e^{x}=1+{x \over 1!}+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}"> Because this series is <a href="/wiki/Convergent_series" title="Convergent series">convergent</a> for every <a href="/wiki/Complex_number" title="Complex number">complex</a> value of x, it is commonly used to extend the definition of ex to the complex numbers.<a href="#cite_note-Dennery-46">[46]</a> This, with the Taylor series for <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">sin and cos x</a>, allows one to derive <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}=\cos x+i\sin x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos x+i\sin x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab1fcd1a6db5cc6678bb9cbd871580eeeb86eda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.999ex; height:3.009ex;" alt="{\displaystyle e^{ix}=\cos x+i\sin x,}"> which holds for every complex x.<a href="#cite_note-Dennery-46">[46]</a> The special case with x = <a href="/wiki/Pi" title="Pi">π</a> is <a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi }+1=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\pi }+1=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47665065f6bec7fb83e5b667197b8f93f4ca36da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.736ex; height:3.009ex;" alt="{\displaystyle e^{i\pi }+1=0,}"> which is considered to be an exemplar of <a href="/wiki/Mathematical_beauty" title="Mathematical beauty">mathematical beauty</a> as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem#Transcendence_of_e_and_π" title="Lindemann–Weierstrass theorem">a proof</a> that π is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>, which implies the impossibility of <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a>.<a href="#cite_note-47">[47]</a><a href="#cite_note-48">[48]</a> Moreover, the identity implies that, in the <a href="/wiki/Principal_branch" title="Principal branch">principal branch</a> of the logarithm,<a href="#cite_note-Dennery-46">[46]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(-1)=i\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(-1)=i\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b623d2fa371489f25ae38c0951f25ea4cc0b89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.599ex; height:2.843ex;" alt="{\displaystyle \ln(-1)=i\pi .}"> Furthermore, using the laws for exponentiation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos nx+i\sin nx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos nx+i\sin nx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f870b74e0018cd897a916beaed2e86940840f33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.954ex; height:3.343ex;" alt="{\displaystyle (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos nx+i\sin nx}"> for any integer n, which is <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a>.<a href="#cite_note-Sultan-49">[49]</a> The expressions of cos x and sin x in terms of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> can be deduced from the Taylor series:<a href="#cite_note-Dennery-46">[46]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75fd5c29bc3f6abb90bcc6270e23e26b2a449170" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.127ex; height:5.676ex;" alt="{\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}"> The expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \cos x+i\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \cos x+i\sin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adaab56b7cb1baa86ebcfb9afa09538e6d0630c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.43ex; height:2.343ex;" alt="{\textstyle \cos x+i\sin x}"> is sometimes abbreviated as cis(x).<a href="#cite_note-Sultan-49">[49]</a> <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=17" title="Edit section: Representations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_representations_of_e" title="List of representations of e">List of representations of e</a></div> The number e can be represented in a variety of ways: as an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>, an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a>, a <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a>, or a <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit of a sequence</a>. In addition to the limit and the series given above, there is also the <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fraction</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...,1,2n,1,...],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...,1,2n,1,...],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda10aefcc9d9e0268fb3b0734147f652800e6ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.308ex; height:2.843ex;" alt="{\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...,1,2n,1,...],}"><a href="#cite_note-50">[50]</a><a href="#cite_note-OEIS_continued_fraction-51">[51]</a></dd></dl> which written out looks like <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1bf3c9f2712a7350f765443d5b75a550bd20e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.171ex; width:45.683ex; height:30.343ex;" alt="{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}}.}"></dd></dl> The following infinite product evaluates to e:<a href="#cite_note-Finch-2003-p14-26">[26]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\frac {2}{1}}\left({\frac {4}{3}}\right)^{1/2}\left({\frac {6\cdot 8}{5\cdot 7}}\right)^{1/4}\left({\frac {10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\right)^{1/8}\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>1</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mo>⋅</mo> <mn>8</mn> </mrow> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>7</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mo>⋅</mo> <mn>12</mn> <mo>⋅</mo> <mn>14</mn> <mo>⋅</mo> <mn>16</mn> </mrow> <mrow> <mn>9</mn> <mo>⋅</mo> <mn>11</mn> <mo>⋅</mo> <mn>13</mn> <mo>⋅</mo> <mn>15</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mrow> </msup> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\frac {2}{1}}\left({\frac {4}{3}}\right)^{1/2}\left({\frac {6\cdot 8}{5\cdot 7}}\right)^{1/4}\left({\frac {10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\right)^{1/8}\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49fb901f1db7a1d6b1437a59be4313a58f38729d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.695ex; height:6.676ex;" alt="{\displaystyle e={\frac {2}{1}}\left({\frac {4}{3}}\right)^{1/2}\left({\frac {6\cdot 8}{5\cdot 7}}\right)^{1/4}\left({\frac {10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\right)^{1/8}\cdots .}"> Many other series, sequence, continued fraction, and infinite product representations of e have been proved. <div class="mw-heading mw-heading3"><h3 id="Stochastic_representations">Stochastic representations</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=18" title="Edit section: Stochastic representations">edit</a>]</div> In addition to exact analytical expressions for representation of e, there are stochastic techniques for estimating e. One such approach begins with an infinite sequence of independent random variables X1, X2..., drawn from the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> on [0, 1]. Let V be the least number n such that the sum of the first n observations exceeds 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\min \left\{n\mid X_{1}+X_{2}+\cdots +X_{n}>1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <mi>n</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\min \left\{n\mid X_{1}+X_{2}+\cdots +X_{n}>1\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229adce5af07552aa13c6bfb51e67fdb35169f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.444ex; height:2.843ex;" alt="{\displaystyle V=\min \left\{n\mid X_{1}+X_{2}+\cdots +X_{n}>1\right\}.}"></dd></dl> Then the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of V is e: E(V) = e.<a href="#cite_note-52">[52]</a><a href="#cite_note-53">[53]</a> <div class="mw-heading mw-heading3"><h3 id="Known_digits">Known digits</h3>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=19" title="Edit section: Known digits">edit</a>]</div> The number of known digits of e has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.<a href="#cite_note-54">[54]</a><a href="#cite_note-55">[55]</a> <table class="wikitable" style="margin: 1em auto 1em auto"> <caption>Number of known decimal digits of e </caption> <tbody><tr> <th>Date</th> <th>Decimal digits</th> <th>Computation performed by </th></tr> <tr> <td>1690</td> <td align="right">1</td> <td><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a><a href="#cite_note-Bernoulli,_1690-11">[11]</a> </td></tr> <tr> <td>1714</td> <td align="right">13</td> <td><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a><a href="#cite_note-56">[56]</a> </td></tr> <tr> <td>1748</td> <td align="right">23</td> <td><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a><a href="#cite_note-57">[57]</a> </td></tr> <tr> <td>1853</td> <td align="right">137</td> <td><a href="/wiki/William_Shanks" title="William Shanks">William Shanks</a><a href="#cite_note-58">[58]</a> </td></tr> <tr> <td>1871</td> <td align="right">205</td> <td>William Shanks<a href="#cite_note-59">[59]</a> </td></tr> <tr> <td>1884</td> <td align="right">346</td> <td>J. Marcus Boorman<a href="#cite_note-60">[60]</a> </td></tr> <tr> <td>1949</td> <td align="right">2,010</td> <td><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> (on the <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a>) </td></tr> <tr> <td>1961</td> <td align="right">100,265</td> <td><a href="/wiki/Daniel_Shanks" title="Daniel Shanks">Daniel Shanks</a> and <a href="/wiki/John_Wrench" title="John Wrench">John Wrench</a><a href="#cite_note-We_have_computed_e_on_a_7090_to_100,265D_by_the_obvious_program.-61">[61]</a> </td></tr> <tr> <td>1978</td> <td align="right">116,000</td> <td><a href="/wiki/Steve_Wozniak" title="Steve Wozniak">Steve Wozniak</a> on the <a href="/wiki/Apple_II" title="Apple II">Apple II</a><a href="#cite_note-wozniak198106-62">[62]</a> </td></tr></tbody></table> Since around 2010, the proliferation of modern high-speed <a href="/wiki/Desktop_computer" title="Desktop computer">desktop computers</a> has made it feasible for amateurs to compute trillions of digits of e within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving e to 31,415,926,535,897 (approximately π×1013) digits.<a href="#cite_note-63">[63]</a> <div class="mw-heading mw-heading2"><h2 id="Computing_the_digits">Computing the digits</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=20" title="Edit section: Computing the digits">edit</a>]</div> One way to compute the digits of e is with the series<a href="#cite_note-Finch-2005-64">[64]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <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> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f88e7e4aec44c76ae1ed2e8262f50c1a8e89d98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.265ex; height:7.009ex;" alt="{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}.}"> A faster method involves two recursive functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493f8914158b9bf218a601c58c038c768d47e567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle p(a,b)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129ca5bfe0bbc5b38b982767e75dcca4c084c85e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.14ex; height:2.843ex;" alt="{\displaystyle q(a,b)}">. The functions are defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {p(a,b)}{q(a,b)}}={\begin{cases}{\binom {1}{b}},&{\text{if }}b=a+1{\text{,}}\\{\binom {p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}},&{\text{otherwise, where }}m=\lfloor (a+b)/2\rfloor .\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>1</mn> <mi>b</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise, where </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {p(a,b)}{q(a,b)}}={\begin{cases}{\binom {1}{b}},&{\text{if }}b=a+1{\text{,}}\\{\binom {p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}},&{\text{otherwise, where }}m=\lfloor (a+b)/2\rfloor .\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5c4a907e96300ce2d3c249767051aff5a34621" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.98ex; margin-bottom: -0.192ex; width:70.376ex; height:7.509ex;" alt="{\displaystyle {\binom {p(a,b)}{q(a,b)}}={\begin{cases}{\binom {1}{b}},&{\text{if }}b=a+1{\text{,}}\\{\binom {p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}},&{\text{otherwise, where }}m=\lfloor (a+b)/2\rfloor .\end{cases}}}"> The expression <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {p(0,n)}{q(0,n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {p(0,n)}{q(0,n)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807036816204d7d716441dc6fb61b24944a2132d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.409ex; height:6.509ex;" alt="{\displaystyle 1+{\frac {p(0,n)}{q(0,n)}}}"> produces the nth partial sum of the series above. This method uses <a href="/wiki/Binary_splitting" title="Binary splitting">binary splitting</a> to compute e with fewer single-digit arithmetic operations and thus reduced <a href="/wiki/Bit_complexity" class="mw-redirect" title="Bit complexity">bit complexity</a>. Combining this with <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>-based methods of multiplying integers makes computing the digits very fast.<a href="#cite_note-Finch-2005-64">[64]</a> <div class="mw-heading mw-heading2"><h2 id="In_computer_culture">In computer culture</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=21" title="Edit section: In computer culture">edit</a>]</div> During the emergence of <a href="/wiki/Internet_culture" title="Internet culture">internet culture</a>, individuals and organizations sometimes paid homage to the number e. In an early example, the <a href="/wiki/Computer_scientist" title="Computer scientist">computer scientist</a> <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> let the version numbers of his program <a href="/wiki/Metafont" title="Metafont">Metafont</a> approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth.<a href="#cite_note-65">[65]</a> In another instance, the <a href="/wiki/Initial_public_offering" title="Initial public offering">IPO</a> filing for <a href="/wiki/Google" title="Google">Google</a> in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 <a href="/wiki/USD" class="mw-redirect" title="USD">USD</a>, which is e billion <a href="/wiki/United_States_dollar" title="United States dollar">dollars</a> rounded to the nearest dollar.<a href="#cite_note-66">[66]</a> Google was also responsible for a billboard<a href="#cite_note-67">[67]</a> that appeared in the heart of <a href="/wiki/Silicon_Valley" title="Silicon Valley">Silicon Valley</a>, and later in <a href="/wiki/Cambridge,_Massachusetts" title="Cambridge, Massachusetts">Cambridge, Massachusetts</a>; <a href="/wiki/Seattle,_Washington" class="mw-redirect" title="Seattle, Washington">Seattle, Washington</a>; and <a href="/wiki/Austin,_Texas" title="Austin, Texas">Austin, Texas</a>. It read "{first 10-digit prime found in consecutive digits of e}.com". The first 10-digit prime in e is 7427466391, which starts at the 99th digit.<a href="#cite_note-68">[68]</a> Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted in finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. It turned out that the sequence consisted of 10-digit numbers found in consecutive digits of e whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit.<a href="#cite_note-69">[69]</a> Solving this second problem finally led to a <a href="/wiki/Google_Labs" title="Google Labs">Google Labs</a> webpage where the visitor was invited to submit a résumé.<a href="#cite_note-70">[70]</a> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=22" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-OEIS_decimal_expansion-1">^ <a href="#cite_ref-OEIS_decimal_expansion_1-0">a</a> <a href="#cite_ref-OEIS_decimal_expansion_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSloane_"A001113"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A001113">"Sequence A001113 (Decimal expansion of e)"</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA001113%26%23x20%3B%28Decimal+expansion+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Ee%3C%2Fspan%3E%29&rft_id=https%3A%2F%2Foeis.org%2FA001113&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Miller-2">^ <a href="#cite_ref-Miller_2-0">a</a> <a href="#cite_ref-Miller_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller" class="citation web cs1">Miller, Jeff. <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/">"Earliest Uses of Symbols for Constants"</a>. MacTutor. University of St. Andrews, Scotland. Retrieved 31 October 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor&rft.atitle=Earliest+Uses+of+Symbols+for+Constants&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FMiller%2Fmathsym%2Fconstants%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Weisstein-3">^ <a href="#cite_ref-Weisstein_3-0">a</a> <a href="#cite_ref-Weisstein_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="mathworld" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/e.html">"e"</a>. mathworld.wolfram.com. Retrieved 2020-08-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=e&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2Fe.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Pickover-4"><a href="#cite_ref-Pickover_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover2009" class="citation book cs1">Pickover, Clifford A. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JrslMKTgSZwC">The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics</a> (illustrated ed.). Sterling Publishing Company. p. 166. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4027-5796-9" title="Special:BookSources/978-1-4027-5796-9"><bdi>978-1-4027-5796-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Math+Book%3A+From+Pythagoras+to+the+57th+Dimension%2C+250+Milestones+in+the+History+of+Mathematics&rft.pages=166&rft.edition=illustrated&rft.pub=Sterling+Publishing+Company&rft.date=2009&rft.isbn=978-1-4027-5796-9&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJrslMKTgSZwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JrslMKTgSZwC&pg=PA166">Extract of page 166</a> </li> <li id="cite_note-OConnor-5">^ <a href="#cite_ref-OConnor_5-0">a</a> <a href="#cite_ref-OConnor_5-1">b</a> <a href="#cite_ref-OConnor_5-2">c</a> <a href="#cite_ref-OConnor_5-3">d</a> <a href="#cite_ref-OConnor_5-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2001" class="citation cs1">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> (September 2001). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/e.html">"The number e"</a>. <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+number+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Ee%3C%2Fspan%3E&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.date=2001-09&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2Fe.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSawyer1961" class="citation book cs1">Sawyer, W. W. (1961). <a rel="nofollow" class="external text" href="https://archive.org/details/MathemateciansDelight-W.W.Sawyer/page/n153/mode/2up">Mathematician's Delight</a>. Penguin. p. 155.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematician%27s+Delight&rft.pages=155&rft.pub=Penguin&rft.date=1961&rft.aulast=Sawyer&rft.aufirst=W.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FMathemateciansDelight-W.W.Sawyer%2Fpage%2Fn153%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2018" class="citation book cs1">Wilson, Robinn (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=345HDwAAQBAJ">Euler's Pioneering Equation: The most beautiful theorem in mathematics</a> (illustrated ed.). Oxford University Press. p. (preface). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-251405-9" title="Special:BookSources/978-0-19-251405-9"><bdi>978-0-19-251405-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler%27s+Pioneering+Equation%3A+The+most+beautiful+theorem+in+mathematics&rft.pages=%28preface%29&rft.edition=illustrated&rft.pub=Oxford+University+Press&rft.date=2018&rft.isbn=978-0-19-251405-9&rft.aulast=Wilson&rft.aufirst=Robinn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D345HDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPosamentierLehmann2004" class="citation book cs1">Posamentier, Alfred S.; Lehmann, Ingmar (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QFPvAAAAMAAJ">Pi: A Biography of the World's Most Mysterious Number</a> (illustrated ed.). Prometheus Books. p. 68. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59102-200-8" title="Special:BookSources/978-1-59102-200-8"><bdi>978-1-59102-200-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi%3A+A+Biography+of+the+World%27s+Most+Mysterious+Number&rft.pages=68&rft.edition=illustrated&rft.pub=Prometheus+Books&rft.date=2004&rft.isbn=978-1-59102-200-8&rft.aulast=Posamentier&rft.aufirst=Alfred+S.&rft.au=Lehmann%2C+Ingmar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQFPvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlverLozierBoisvertClark2010" class="citation cs2"><a href="/wiki/Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/">"E (mathematical constant)"</a>, <a href="/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19225-5" title="Special:BookSources/978-0-521-19225-5"><bdi>978-0-521-19225-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=E+%28mathematical+constant%29&rft.btitle=NIST+Handbook+of+Mathematical+Functions&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-19225-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2723248%23id-name%3DMR&rft_id=http%3A%2F%2Fdlmf.nist.gov%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988">. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruins1983" class="citation journal cs1">Bruins, E. M. (1983). <a rel="nofollow" class="external text" href="http://www.math.bas.bg/mathmod/Proceedings_CTF/CTF-1981/files_CTF-1981/CTF-1981-254-257.pdf">"The Computation of Logarithms by Huygens"</a> (PDF). Constructive Function Theory: 254–257.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Constructive+Function+Theory&rft.atitle=The+Computation+of+Logarithms+by+Huygens&rft.pages=254-257&rft.date=1983&rft.aulast=Bruins&rft.aufirst=E.+M.&rft_id=http%3A%2F%2Fwww.math.bas.bg%2Fmathmod%2FProceedings_CTF%2FCTF-1981%2Ffiles_CTF-1981%2FCTF-1981-254-257.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Bernoulli,_1690-11">^ <a href="#cite_ref-Bernoulli,_1690_11-0">a</a> <a href="#cite_ref-Bernoulli,_1690_11-1">b</a> Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for e. See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–23. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222">On page 222</a>, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would be owing [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si a = b, debebitur plu quam 2½a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) If a = b, the geometric series reduces to the series for a × e, so 2.5 < e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k56536t/f307.image.langEN">page 314.</a>) </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarl_BoyerUta_Merzbach1991" class="citation book cs1">Carl Boyer; <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Uta Merzbach</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2nd ed.). Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/419">419</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=419&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.au=Carl+Boyer&rft.au=Uta+Merzbach&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeibniz2003" class="citation web cs1 cs1-prop-foreign-lang-source">Leibniz, Gottfried Wilhelm (2003). <a rel="nofollow" class="external text" href="https://leibniz.uni-goettingen.de/files/pdf/Leibniz-Edition-III-5.pdf">"Sämliche Schriften Und Briefe"</a> (PDF) (in German). <q>look for example letter nr. 6</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=S%C3%A4mliche+Schriften+Und+Briefe&rft.date=2003&rft.aulast=Leibniz&rft.aufirst=Gottfried+Wilhelm&rft_id=https%3A%2F%2Fleibniz.uni-goettingen.de%2Ffiles%2Fpdf%2FLeibniz-Edition-III-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Meditatio-14"><a href="#cite_ref-Meditatio_14-0">^</a> Euler, <a rel="nofollow" class="external text" href="https://scholarlycommons.pacific.edu/euler-works/853/">Meditatio in experimenta explosione tormentorum nuper instituta</a>. Scribatur pro numero cujus logarithmus est unitas, e, qui est 2,7182817… (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...") </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60, see especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gf1OEXIQQgsC&pg=PA58">p. 58.</a> From p. 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … ) </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRemmert1991" class="citation book cs1"><a href="/wiki/Reinhold_Remmert" title="Reinhold Remmert">Remmert, Reinhold</a> (1991). Theory of Complex Functions. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 136. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97195-7" title="Special:BookSources/978-0-387-97195-7"><bdi>978-0-387-97195-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Complex+Functions&rft.pages=136&rft.pub=Springer-Verlag&rft.date=1991&rft.isbn=978-0-387-97195-7&rft.aulast=Remmert&rft.aufirst=Reinhold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qalsP7uMiV4C&pg=PA68">From page 68:</a> Erit enim <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>c</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>r</mi> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5f6a94a29fc0aed56962f557ad1cf5ca540615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.671ex; height:5.509ex;" alt="{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}"> seu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>r</mi> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69734b7fb10c894953c8a773dac844cdb1004cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.332ex; height:3.843ex;" alt="{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}"> ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>c</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>r</mi> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5f6a94a29fc0aed56962f557ad1cf5ca540615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.671ex; height:5.509ex;" alt="{\displaystyle {\frac {dc}{c}}={\frac {dyds}{rdx}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>r</mi> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69734b7fb10c894953c8a773dac844cdb1004cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.332ex; height:3.843ex;" alt="{\displaystyle c=e^{\int {\frac {dyds}{rdx}}}}">, where e denotes the number whose hyperbolic [i.e., natural] logarithm is 1.) </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCalinger2016" class="citation book cs1">Calinger, Ronald (2016). Leonhard Euler: Mathematical Genius in the Enlightenment. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11927-4" title="Special:BookSources/978-0-691-11927-4"><bdi>978-0-691-11927-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Leonhard+Euler%3A+Mathematical+Genius+in+the+Enlightenment&rft.pub=Princeton+University+Press&rft.date=2016&rft.isbn=978-0-691-11927-4&rft.aulast=Calinger&rft.aufirst=Ronald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> p. 124. </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). Principles of Mathematical Analysis (3rd ed.). 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William Morrow. pp. 29–32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-06-168909-3" title="Special:BookSources/978-0-06-168909-3"><bdi>978-0-06-168909-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cartoon+Guide+to+Calculus&rft.pages=29-32&rft.pub=William+Morrow&rft.date=2012&rft.isbn=978-0-06-168909-3&rft.aulast=Gonick&rft.aufirst=Larry&rft_id=https%3A%2F%2Fwww.larrygonick.com%2Ftitles%2Fscience%2Fcartoon-guide-to-calculus-2%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-:0-21">^ <a href="#cite_ref-:0_21-0">a</a> <a href="#cite_ref-:0_21-1">b</a> <a href="#cite_ref-:0_21-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramson2023" class="citation book cs1">Abramson, Jay; et al. 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Archived from <a rel="nofollow" class="external text" href="http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html">the original</a> on 2011-07-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability&rft.pages=85&rft.pub=American+Mathematical+Society&rft.date=1997&rft.isbn=978-0-8218-9414-9&rft.aulast=Grinstead&rft.aufirst=Charles+M.&rft.au=Snell%2C+James+Laurie&rft_id=http%3A%2F%2Fwww.dartmouth.edu%2F~chance%2Fteaching_aids%2Fbooks_articles%2Fprobability_book%2Fbook.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. 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Addison-Wesley. p. 183. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-03801-3" title="Special:BookSources/0-201-03801-3"><bdi>0-201-03801-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.pages=183&rft.pub=Addison-Wesley&rft.date=1997&rft.isbn=0-201-03801-3&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Finch-2003-p14-26">^ <a href="#cite_ref-Finch-2003-p14_26-0">a</a> <a href="#cite_ref-Finch-2003-p14_26-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteven_Finch2003" class="citation book cs1">Steven Finch (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalcons0000finc">Mathematical constants</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalcons0000finc/page/14">14</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-81805-6" title="Special:BookSources/978-0-521-81805-6"><bdi>978-0-521-81805-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+constants&rft.pages=14&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-81805-6&rft.au=Steven+Finch&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalcons0000finc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-greg-27">^ <a href="#cite_ref-greg_27-0">a</a> <a href="#cite_ref-greg_27-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGbur2011" class="citation book cs1"><a href="/wiki/Greg_Gbur" title="Greg Gbur">Gbur, Greg</a> (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. p. 779. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521516-10-5" title="Special:BookSources/978-0-521516-10-5"><bdi>978-0-521516-10-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Optical+Physics+and+Engineering&rft.pages=779&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-0-521516-10-5&rft.aulast=Gbur&rft.aufirst=Greg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-kline-28">^ <a href="#cite_ref-kline_28-0">a</a> <a href="#cite_ref-kline_28-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1998" class="citation book cs1">Kline, M. (1998). 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OpenStax. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-947172-14-2" title="Special:BookSources/978-1-947172-14-2"><bdi>978-1-947172-14-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6.3+Taylor+and+Maclaurin+Series&rft.btitle=Calculus%2C+volume+2&rft.pub=OpenStax&rft.date=2023&rft.isbn=978-1-947172-14-2&rft.aulast=Strang&rft.aufirst=Gilbert&rft.au=Herman%2C+Edwin&rft_id=https%3A%2F%2Fopenstax.org%2Fbooks%2Fcalculus-volume-2%2Fpages%2F6-3-taylor-and-maclaurin-series&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrangHerman2023" class="citation book cs1">Strang, Gilbert; Herman, Edwin; et al. (2023). <a rel="nofollow" class="external text" href="https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives">"4.10 Antiderivatives"</a>. Calculus, volume 2. 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The American Statistician. 45 (1): 66–68. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1991.10475769">10.1080/00031305.1991.10475769</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2685243">2685243</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Estimating+the+Value+of+e+by+Simulation&rft.volume=45&rft.issue=1&rft.pages=66-68&rft.date=1991-02&rft_id=info%3Adoi%2F10.1080%2F00031305.1991.10475769&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2685243%23id-name%3DJSTOR&rft.aulast=Russell&rft.aufirst=K.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Dinov, ID (2007) <a rel="nofollow" class="external text" href="http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_e_using_SOCR_simulation">Estimating e using SOCR simulation</a>, SOCR Hands-on Activities (retrieved December 26, 2007). </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> Sebah, P. and Gourdon, X.; <a rel="nofollow" class="external text" href="http://numbers.computation.free.fr/Constants/E/e.html">The constant e and its computation</a> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Gourdon, X.; <a rel="nofollow" class="external text" href="http://numbers.computation.free.fr/Constants/PiProgram/computations.html">Reported large computations with PiFast</a> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5–45; <a rel="nofollow" class="external text" href="https://archive.today/20140410203227/http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=16">see especially the bottom of page 10.</a> From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … ) </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_jQ1bAAAAQAAJ/page/n115">page 90.</a> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> William Shanks, Contributions to Mathematics, ... (London, England: G. Bell, 1853), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d-9ZAAAAcAAJ&pg=PA89">page 89.</a> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> William Shanks (1871) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sclTAAAAcAAJ&pg=PA27">"On the numerical values of e, loge 2, loge 3, loge 5, and loge 10, also on the numerical value of M the modulus of the common system of logarithms, all to 205 decimals,"</a> Proceedings of the Royal Society of London, 20 : 27–29. </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> J. Marcus Boorman (October 1884) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mG8yAQAAMAAJ&pg=PA204">"Computation of the Naperian base,"</a> Mathematical Magazine, 1 (12) : 204–205. </li> <li id="cite_note-We_have_computed_e_on_a_7090_to_100,265D_by_the_obvious_program.-61"><a href="#cite_ref-We_have_computed_e_on_a_7090_to_100,265D_by_the_obvious_program._61-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaniel_ShanksJohn_W_Wrench1962" class="citation journal cs1"><a href="/wiki/Daniel_Shanks" title="Daniel Shanks">Daniel Shanks</a>; <a href="/wiki/John_Wrench" title="John Wrench">John W Wrench</a> (1962). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1962-16-077/S0025-5718-1962-0136051-9/S0025-5718-1962-0136051-9.pdf">"Calculation of Pi to 100,000 Decimals"</a> (PDF). Mathematics of Computation. 16 (77): 76–99. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2003813">10.2307/2003813</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2003813">2003813</a>. p. 78: <q>We have computed e on a 7090 to 100,265D by the obvious program</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Calculation+of+Pi+to+100%2C000+Decimals&rft.volume=16&rft.issue=77&rft.pages=76-99&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F2003813&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2003813%23id-name%3DJSTOR&rft.au=Daniel+Shanks&rft.au=John+W+Wrench&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1962-16-077%2FS0025-5718-1962-0136051-9%2FS0025-5718-1962-0136051-9.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-wozniak198106-62"><a href="#cite_ref-wozniak198106_62-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWozniak1981" class="citation magazine cs1"><a href="/wiki/Steve_Wozniak" title="Steve Wozniak">Wozniak, Steve</a> (June 1981). <a rel="nofollow" class="external text" href="https://archive.org/stream/byte-magazine-1981-06/1981_06_BYTE_06-06_Operating_Systems#page/n393/mode/2up">"The Impossible Dream: Computing e to 116,000 Places with a Personal Computer"</a>. BYTE. Vol. 6, no. 6. McGraw-Hill. p. 392. Retrieved 18 October 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BYTE&rft.atitle=The+Impossible+Dream%3A+Computing+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Ee%3C%2Fspan%3E+to+116%2C000+Places+with+a+Personal+Computer&rft.volume=6&rft.issue=6&rft.pages=392&rft.date=1981-06&rft.aulast=Wozniak&rft.aufirst=Steve&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fbyte-magazine-1981-06%2F1981_06_BYTE_06-06_Operating_Systems%23page%2Fn393%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_Yee2020" class="citation web cs1">Alexander Yee, ed. (5 December 2020). <a rel="nofollow" class="external text" href="http://www.numberworld.org/digits/E/">"e"</a>. Numberworld.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberworld&rft.atitle=e&rft.date=2020-12-05&rft_id=http%3A%2F%2Fwww.numberworld.org%2Fdigits%2FE%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-Finch-2005-64">^ <a href="#cite_ref-Finch-2005_64-0">a</a> <a href="#cite_ref-Finch-2005_64-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFinch2005" class="citation book cs1">Finch, Steven R. (2005). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/180072364">Mathematical constants</a>. Cambridge Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-81805-6" title="Special:BookSources/978-0-521-81805-6"><bdi>978-0-521-81805-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/180072364">180072364</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+constants&rft.pub=Cambridge+Univ.+Press&rft.date=2005&rft_id=info%3Aoclcnum%2F180072364&rft.isbn=978-0-521-81805-6&rft.aulast=Finch&rft.aufirst=Steven+R.&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F180072364&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1990" class="citation journal cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1990-10-03). <a rel="nofollow" class="external text" href="http://www.ntg.nl/maps/05/34.pdf">"The Future of TeX and Metafont"</a> (PDF). TeX Mag. 5 (1): 145. Retrieved 2017-02-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=TeX+Mag&rft.atitle=The+Future+of+TeX+and+Metafont&rft.volume=5&rft.issue=1&rft.pages=145&rft.date=1990-10-03&rft.aulast=Knuth&rft.aufirst=Donald&rft_id=http%3A%2F%2Fwww.ntg.nl%2Fmaps%2F05%2F34.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobergeMelançon2017" class="citation journal cs1">Roberge, Jonathan; Melançon, Louis (June 2017). <a rel="nofollow" class="external text" href="http://journals.sagepub.com/doi/10.1177/1354856515592506">"Being the King Kong of algorithmic culture is a tough job after all: Google's regimes of justification and the meanings of Glass"</a>. Convergence: The International Journal of Research into New Media Technologies. 23 (3): 306–324. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F1354856515592506">10.1177/1354856515592506</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1354-8565">1354-8565</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Convergence%3A+The+International+Journal+of+Research+into+New+Media+Technologies&rft.atitle=Being+the+King+Kong+of+algorithmic+culture+is+a+tough+job+after+all%3A+Google%27s+regimes+of+justification+and+the+meanings+of+Glass&rft.volume=23&rft.issue=3&rft.pages=306-324&rft.date=2017-06&rft_id=info%3Adoi%2F10.1177%2F1354856515592506&rft.issn=1354-8565&rft.aulast=Roberge&rft.aufirst=Jonathan&rft.au=Melan%C3%A7on%2C+Louis&rft_id=http%3A%2F%2Fjournals.sagepub.com%2Fdoi%2F10.1177%2F1354856515592506&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131203102744/http://braintags.com/archives/2004/07/first-10digit-prime-found-in-consecutive-digits-of-e/">"First 10-digit prime found in consecutive digits of e"</a>. Brain Tags. Archived from <a rel="nofollow" class="external text" href="http://braintags.com/archives/2004/07/first-10digit-prime-found-in-consecutive-digits-of-e/">the original</a> on 2013-12-03. Retrieved 2012-02-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Brain+Tags&rft.atitle=First+10-digit+prime+found+in+consecutive+digits+of+%3Cspan+class%3D%22texhtml+%22+%3Ee%3C%2Fspan%3E&rft_id=http%3A%2F%2Fbraintags.com%2Farchives%2F2004%2F07%2Ffirst-10digit-prime-found-in-consecutive-digits-of-e%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKazmierczak2004" class="citation web cs1">Kazmierczak, Marcus (2004-07-29). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100923111259/http://mkaz.com/math/google-billboard/">"Google Billboard"</a>. mkaz.com. Archived from <a rel="nofollow" class="external text" href="http://mkaz.com/math/google-billboard">the original</a> on 2010-09-23. Retrieved 2007-06-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Google+Billboard&rft.pub=mkaz.com&rft.date=2004-07-29&rft.aulast=Kazmierczak&rft.aufirst=Marcus&rft_id=http%3A%2F%2Fmkaz.com%2Fmath%2Fgoogle-billboard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a rel="nofollow" class="external text" href="http://explorepdx.com/firsten.html">The first 10-digit prime in e</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210411010312/http://explorepdx.com/firsten.html">Archived</a> 2021-04-11 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Explore Portland Community. Retrieved on 2020-12-09. </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShea" class="citation news cs1">Shea, Andrea. <a rel="nofollow" class="external text" href="https://www.npr.org/templates/story/story.php?storyId=3916173">"Google Entices Job-Searchers with Math Puzzle"</a>. NPR. Retrieved 2007-06-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NPR&rft.atitle=Google+Entices+Job-Searchers+with+Math+Puzzle&rft.aulast=Shea&rft.aufirst=Andrea&rft_id=https%3A%2F%2Fwww.npr.org%2Ftemplates%2Fstory%2Fstory.php%3FstoryId%3D3916173&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=23" title="Edit section: Further reading">edit</a>]</div> <ul><li>Maor, Eli; e: The Story of a Number, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-05854-7" title="Special:BookSources/0-691-05854-7">0-691-05854-7</a></li> <li><a rel="nofollow" class="external text" href="http://www.johnderbyshire.com/Books/Prime/Blog/page.html#endnote10">Commentary on Endnote 10</a> of the book <a href="/wiki/Prime_Obsession" title="Prime Obsession">Prime Obsession</a> for another stochastic representation</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCartin,_Brian_J.2006" class="citation journal cs1">McCartin, Brian J. (2006). <a rel="nofollow" class="external text" href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf">"e: The Master of All"</a> (PDF). The Mathematical Intelligencer. 28 (2): 10–21. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02987150">10.1007/bf02987150</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123033482">123033482</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=e%3A+The+Master+of+All&rft.volume=28&rft.issue=2&rft.pages=10-21&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2Fbf02987150&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123033482%23id-name%3DS2CID&rft.au=McCartin%2C+Brian+J.&rft_id=http%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fupload_library%2F22%2FChauvenet%2Fmccartin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE+%28mathematical+constant%29" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=E_(mathematical_constant)&action=edit&section=24" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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2008</li> <li><a rel="nofollow" class="external text" href="http://www.gresham.ac.uk/lectures-and-events/the-story-of-e">"The story of e"</a>, by Robin Wilson at <a href="/wiki/Gresham_College" title="Gresham College">Gresham College</a>, 28 February 2007 (available for audio and video download)</li> <li><a rel="nofollow" class="external text" href="http://pisearch.org/e">e Search Engine</a> 2 billion searchable digits of e, π and √2</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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.navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Irrational_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Irrational_number" title="Template:Irrational number"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Irrational_number" title="Template talk:Irrational number"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Irrational_number" title="Special:EditPage/Template:Irrational number"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Irrational_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chaitin%27s_constant" title="Chaitin's constant">Chaitin's</a> (Ω)</li> <li><a href="/wiki/Liouville_number" title="Liouville number">Liouville</a></li> <li><a href="/wiki/Prime_constant" title="Prime constant">Prime</a> (ρ)</li> <li><a href="/wiki/Omega_constant" title="Omega constant">Omega</a></li> <li><a href="/wiki/Cahen%27s_constant" title="Cahen's constant">Cahen</a></li></ul> <ul><li><a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">Logarithm of 2</a></li> <li><a href="/wiki/Dottie_number" title="Dottie number">Dottie</a></li> <li><a href="/wiki/Lemniscate_constant" title="Lemniscate constant">Lemniscate</a> (ϖ)</li> <li><a href="/wiki/Twelfth_root_of_two" title="Twelfth root of two">Twelfth root of 2</a></li> <li><a href="/wiki/Ap%C3%A9ry%27s_constant" title="Apéry's constant">Apéry's</a> (ζ(3))</li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Cube root of 2</a></li> <li><a href="/wiki/Plastic_ratio" title="Plastic ratio">Plastic ratio</a> (ρ)</li></ul> <ul><li><a href="/wiki/Square_root_of_2" title="Square root of 2">Square root of 2</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio</a> (ψ)</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Borwein_constant" title="Erdős–Borwein constant">Erdős–Borwein</a> (E)</li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a> (φ)</li> <li><a href="/wiki/Square_root_of_3" title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio</a> (ς)</li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver ratio</a> (δS)</li> <li><a href="/wiki/Square_root_of_6" title="Square root of 6">Square root of 6</a></li> <li><a href="/wiki/Square_root_of_7" title="Square root of 7">Square root of 7</a></li></ul> <ul><li><a class="mw-selflink selflink">Euler's</a> (e)</li> <li><a href="/wiki/Pi" title="Pi">Pi</a> (π)</li></ul> </div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Gold,_square_root_of_2,_and_square_root_of_3_rectangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/50px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png" decoding="async" width="50" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/75px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/100px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png 2x" 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e (mathematical constant) - Wikipedia