Pi - Wikipedia

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cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Polygon_approximation_era" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polygon_approximation_era"> <div class="vector-toc-text"> 2.1 Polygon approximation era </div> </a> <ul id="toc-Polygon_approximation_era-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_series"> <div class="vector-toc-text"> 2.2 Infinite series </div> </a> <ul id="toc-Infinite_series-sublist" class="vector-toc-list"> <li id="toc-Rate_of_convergence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rate_of_convergence"> <div class="vector-toc-text"> 2.2.1 Rate of convergence </div> </a> <ul id="toc-Rate_of_convergence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Irrationality_and_transcendence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irrationality_and_transcendence"> <div class="vector-toc-text"> 2.3 Irrationality and transcendence </div> </a> <ul id="toc-Irrationality_and_transcendence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adoption_of_the_symbol_π" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adoption_of_the_symbol_π"> <div class="vector-toc-text"> 2.4 Adoption of the symbol π </div> </a> <ul id="toc-Adoption_of_the_symbol_π-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modern_quest_for_more_digits" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Modern_quest_for_more_digits"> <div class="vector-toc-text"> 3 Modern quest for more digits </div> </a> <button aria-controls="toc-Modern_quest_for_more_digits-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Modern quest for more digits subsection </button> <ul id="toc-Modern_quest_for_more_digits-sublist" class="vector-toc-list"> <li id="toc-Computer_era_and_iterative_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_era_and_iterative_algorithms"> <div class="vector-toc-text"> 3.1 Computer era and iterative algorithms </div> </a> <ul id="toc-Computer_era_and_iterative_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Motives_for_computing_π" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motives_for_computing_π"> <div class="vector-toc-text"> 3.2 Motives for computing π </div> </a> <ul id="toc-Motives_for_computing_π-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rapidly_convergent_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rapidly_convergent_series"> <div class="vector-toc-text"> 3.3 Rapidly convergent series </div> </a> <ul id="toc-Rapidly_convergent_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monte_Carlo_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Monte_Carlo_methods"> <div class="vector-toc-text"> 3.4 Monte Carlo methods </div> </a> <ul id="toc-Monte_Carlo_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spigot_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spigot_algorithms"> <div class="vector-toc-text"> 3.5 Spigot algorithms </div> </a> <ul id="toc-Spigot_algorithms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Role_and_characterizations_in_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Role_and_characterizations_in_mathematics"> <div class="vector-toc-text"> 4 Role and characterizations in mathematics </div> </a> <button aria-controls="toc-Role_and_characterizations_in_mathematics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Role and characterizations in mathematics subsection </button> <ul id="toc-Role_and_characterizations_in_mathematics-sublist" class="vector-toc-list"> <li id="toc-Geometry_and_trigonometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_and_trigonometry"> <div class="vector-toc-text"> 4.1 Geometry and trigonometry </div> </a> <ul id="toc-Geometry_and_trigonometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Units_of_angle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Units_of_angle"> <div class="vector-toc-text"> 4.2 Units of angle </div> </a> <ul id="toc-Units_of_angle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenvalues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvalues"> <div class="vector-toc-text"> 4.3 Eigenvalues </div> </a> <ul id="toc-Eigenvalues-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"> 4.4 Inequalities </div> </a> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_transform_and_Heisenberg_uncertainty_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_transform_and_Heisenberg_uncertainty_principle"> <div class="vector-toc-text"> 4.5 Fourier transform and Heisenberg uncertainty principle </div> </a> <ul id="toc-Fourier_transform_and_Heisenberg_uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussian_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gaussian_integrals"> <div class="vector-toc-text"> 4.6 Gaussian integrals </div> </a> <ul id="toc-Gaussian_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> 4.7 Topology </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cauchy's_integral_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cauchy's_integral_formula"> <div class="vector-toc-text"> 4.8 Cauchy's integral formula </div> </a> <ul id="toc-Cauchy's_integral_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_calculus_and_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_calculus_and_physics"> <div class="vector-toc-text"> 4.9 Vector calculus and physics </div> </a> <ul id="toc-Vector_calculus_and_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_curvature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_curvature"> <div class="vector-toc-text"> 4.10 Total curvature </div> </a> <ul id="toc-Total_curvature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_gamma_function_and_Stirling's_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_gamma_function_and_Stirling's_approximation"> <div class="vector-toc-text"> 4.11 The gamma function and Stirling's approximation </div> </a> <ul id="toc-The_gamma_function_and_Stirling's_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory_and_Riemann_zeta_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory_and_Riemann_zeta_function"> <div class="vector-toc-text"> 4.12 Number theory and Riemann zeta function </div> </a> <ul id="toc-Number_theory_and_Riemann_zeta_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_series"> <div class="vector-toc-text"> 4.13 Fourier series </div> </a> <ul id="toc-Fourier_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modular_forms_and_theta_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modular_forms_and_theta_functions"> <div class="vector-toc-text"> 4.14 Modular forms and theta functions </div> </a> <ul id="toc-Modular_forms_and_theta_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cauchy_distribution_and_potential_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cauchy_distribution_and_potential_theory"> <div class="vector-toc-text"> 4.15 Cauchy distribution and potential theory </div> </a> <ul id="toc-Cauchy_distribution_and_potential_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_Mandelbrot_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_the_Mandelbrot_set"> <div class="vector-toc-text"> 4.16 In the Mandelbrot set </div> </a> <ul id="toc-In_the_Mandelbrot_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_geometry"> <div class="vector-toc-text"> 4.17 Projective geometry </div> </a> <ul id="toc-Projective_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Outside_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Outside_mathematics"> <div class="vector-toc-text"> 5 Outside mathematics </div> </a> <button aria-controls="toc-Outside_mathematics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Outside mathematics subsection </button> <ul id="toc-Outside_mathematics-sublist" class="vector-toc-list"> <li id="toc-Describing_physical_phenomena" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Describing_physical_phenomena"> <div class="vector-toc-text"> 5.1 Describing physical phenomena </div> </a> <ul id="toc-Describing_physical_phenomena-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Memorizing_digits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Memorizing_digits"> <div class="vector-toc-text"> 5.2 Memorizing digits </div> </a> <ul id="toc-Memorizing_digits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> 5.3 In popular culture </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 6 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 7 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Explanatory_notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Explanatory_notes"> <div class="vector-toc-text"> 7.1 Explanatory notes </div> </a> <ul id="toc-Explanatory_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 7.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_and_cited_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_and_cited_sources"> <div class="vector-toc-text"> 7.3 General and cited sources </div> </a> <ul id="toc-General_and_cited_sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 8 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"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 9 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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Pi</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 162 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-162" aria-hidden="true" > 162 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://af.wikipedia.org/wiki/Pi" title="Pi – Afrikaans" lang="af" hreflang="af" data-title="Pi" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Pi_(Mathematik)" title="Pi (Mathematik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Pi (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%93%E1%8B%AD" title="ፓይ – Amharic" lang="am" hreflang="am" data-title="ፓይ" 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/%D8%B7_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="ط (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="ط (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_%CF%80" title="Numero π – Aragonese" lang="an" hreflang="an" data-title="Numero π" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%87" title="পাই – Assamese" lang="as" hreflang="as" data-title="পাই" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_%CF%80" title="Númberu π – Asturian" lang="ast" hreflang="ast" data-title="Númberu π" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Pi" title="Pi – Guarani" lang="gn" hreflang="gn" data-title="Pi" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Pi" title="Pi – Azerbaijani" lang="az" hreflang="az" data-title="Pi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%BE%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%E2%80%8C%D8%B3%DB%8C" title="پی سایی‌سی – South Azerbaijani" lang="azb" hreflang="azb" data-title="پی سایی‌سی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%87" title="পাই – Bangla" lang="bn" hreflang="bn" data-title="পাই" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Pi" title="Pi – Banjar" lang="bjn" hreflang="bjn" data-title="Pi" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target">Banjar</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/%C3%8E%E2%81%BF-chiu-lu%CC%8Dt" title="Îⁿ-chiu-lu̍t – Minnan" lang="nan" hreflang="nan" data-title="Îⁿ-chiu-lu̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9F%D0%B8_(%D2%BB%D0%B0%D0%BD)" title="Пи (һан) – Bashkir" lang="ba" hreflang="ba" data-title="Пи (һан)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%96" title="Пі – Belarusian" lang="be" hreflang="be" data-title="Пі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D1%96" title="Пі – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Пі" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Pi" title="Pi – Central Bikol" lang="bcl" hreflang="bcl" data-title="Pi" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Bulgarian" lang="bg" hreflang="bg" data-title="Пи" 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/Pi" title="Pi – Bosnian" lang="bs" hreflang="bs" data-title="Pi" 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/Pi_(niver)" title="Pi (niver) – Breton" lang="br" hreflang="br" data-title="Pi (niver)" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%82%D0%BE%D0%BE)" title="Пи (тоо) – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Пи (тоо)" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Nombre_%CF%80" title="Nombre π – Catalan" lang="ca" hreflang="ca" data-title="Nombre π" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%85%D0%B8%D1%81%D0%B5%D0%BF)" title="Пи (хисеп) – Chuvash" lang="cv" hreflang="cv" data-title="Пи (хисеп)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Pi" title="Pi – Cebuano" lang="ceb" hreflang="ceb" data-title="Pi" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://cs.wikipedia.org/wiki/P%C3%AD_(%C4%8D%C3%ADslo)" title="Pí (číslo) – Czech" lang="cs" hreflang="cs" data-title="Pí (číslo)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pi_(mathemateg)" title="Pi (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Pi (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pi_(tal)" title="Pi (tal) – Danish" lang="da" hreflang="da" data-title="Pi (tal)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl – German" lang="de" hreflang="de" data-title="Kreiszahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Konstanta_%CF%80" title="Konstanta π – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Konstanta π" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pii" title="Pii – Estonian" lang="et" hreflang="et" data-title="Pii" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%84%CE%B1%CE%B8%CE%B5%CF%81%CE%AC)" title="Π (μαθηματική σταθερά) – Greek" lang="el" hreflang="el" data-title="Π (μαθηματική σταθερά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Pi_gr%C4%93c" title="Pi grēc – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Pi grēc" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_%CF%80" title="Número π – Spanish" lang="es" hreflang="es" data-title="Número π" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://eo.wikipedia.org/wiki/Pi_(nombro)" title="Pi (nombro) – Esperanto" lang="eo" hreflang="eo" data-title="Pi (nombro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/N%C3%BAmiru_%CF%80" title="Númiru π – Extremaduran" lang="ext" hreflang="ext" data-title="Númiru π" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Pi_(zenbakia)" title="Pi (zenbakia) – Basque" lang="eu" hreflang="eu" data-title="Pi (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/%D8%B9%D8%AF%D8%AF_%D9%BE%DB%8C" title="عدد پی – Persian" lang="fa" hreflang="fa" data-title="عدد پی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Pi" title="Pi – Fiji Hindi" lang="hif" hreflang="hif" data-title="Pi" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Pi" title="Pi – Faroese" lang="fo" hreflang="fo" data-title="Pi" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Pi" title="Pi – French" lang="fr" hreflang="fr" data-title="Pi" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Py_(wiskunde)" title="Py (wiskunde) – Western Frisian" lang="fy" hreflang="fy" data-title="Py (wiskunde)" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Pi_gr%C3%AAc" title="Pi grêc – Friulian" lang="fur" hreflang="fur" data-title="Pi grêc" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target">Furlan</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/P%C3%AD_(uimhir)" title="Pí (uimhir) – Irish" lang="ga" hreflang="ga" data-title="Pí (uimhir)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Pi_(%C3%A0ireamh)" title="Pi (àireamh) – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Pi (àireamh)" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_pi" title="Número pi – Galician" lang="gl" hreflang="gl" data-title="Número pi" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 – Gan" lang="gan" hreflang="gan" data-title="圓周率" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AA%BE%E0%AA%87" title="પાઇ – Gujarati" lang="gu" hreflang="gu" data-title="પાઇ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Kalmyk" lang="xal" hreflang="xal" data-title="Пи" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율 – Korean" lang="ko" hreflang="ko" data-title="원주율" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Pi" title="Pi – Hausa" lang="ha" hreflang="ha" data-title="Pi" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Pai_(makemakika)" title="Pai (makemakika) – Hawaiian" lang="haw" hreflang="haw" data-title="Pai (makemakika)" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AB_%D5%A9%D5%AB%D5%BE" title="Պի թիվ – Armenian" lang="hy" hreflang="hy" data-title="Պի թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%88" title="पाई – Hindi" lang="hi" hreflang="hi" data-title="पाई" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Konstanta_%CF%80" title="Konstanta π – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Konstanta π" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pi_(broj)" title="Pi (broj) – Croatian" lang="hr" hreflang="hr" data-title="Pi (broj)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Pi" title="Pi – Ido" lang="io" hreflang="io" data-title="Pi" 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/Pi" title="Pi – Indonesian" lang="id" hreflang="id" data-title="Pi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Pi" title="Pi – Interlingua" lang="ia" hreflang="ia" data-title="Pi" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Ossetic" lang="os" hreflang="os" data-title="Пи" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Phi" title="Phi – Xhosa" lang="xh" hreflang="xh" data-title="Phi" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/P%C3%AD" title="Pí – Icelandic" lang="is" hreflang="is" data-title="Pí" 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/Pi_greco" title="Pi greco – Italian" lang="it" hreflang="it" data-title="Pi greco" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%90%D7%99" title="פאי – Hebrew" lang="he" hreflang="he" data-title="פאי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pi" title="Pi – Javanese" lang="jv" hreflang="jv" data-title="Pi" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%88" title="ಪೈ – Kannada" lang="kn" hreflang="kn" data-title="ಪೈ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%98_(%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98)" title="პი (რიცხვი) – Georgian" lang="ka" hreflang="ka" data-title="პი (რიცხვი)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%81%D0%B0%D0%BD)" title="Пи (сан) – Kazakh" lang="kk" hreflang="kk" data-title="Пи (сан)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Pi" title="Pi – Cornish" lang="kw" hreflang="kw" data-title="Pi" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Pai" title="Pai – Swahili" lang="sw" hreflang="sw" data-title="Pai" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Pi_(matematik)" title="Pi (matematik) – Haitian Creole" lang="ht" hreflang="ht" data-title="Pi (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Pi" title="Pi – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Pi" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Pi" title="Pi – Kurdish" lang="ku" hreflang="ku" data-title="Pi" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Kyrgyz" lang="ky" hreflang="ky" data-title="Пи" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Numerus_pi" title="Numerus pi – Latin" lang="la" hreflang="la" data-title="Numerus pi" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/P%C4%AB" title="Pī – Latvian" lang="lv" hreflang="lv" data-title="Pī" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Pi_(Zuel)" title="Pi (Zuel) – Luxembourgish" lang="lb" hreflang="lb" data-title="Pi (Zuel)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="Пи (число) – Lezghian" lang="lez" hreflang="lez" data-title="Пи (число)" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target">Лезги</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pi" title="Pi – Lithuanian" lang="lt" hreflang="lt" data-title="Pi" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Pi_(wisk%C3%B3nde)" title="Pi (wiskónde) – Limburgish" lang="li" hreflang="li" data-title="Pi (wiskónde)" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Pi_gregh" title="Pi gregh – Lombard" lang="lmo" hreflang="lmo" data-title="Pi gregh" 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/Pi_(sz%C3%A1m)" title="Pi (szám) – Hungarian" lang="hu" hreflang="hu" data-title="Pi (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%9F%D0%B8" title="Пи – Macedonian" lang="mk" hreflang="mk" data-title="Пи" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Pi" title="Pi – Malagasy" lang="mg" hreflang="mg" data-title="Pi" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%88_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="പൈ (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="പൈ (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%AA%E0%A4%BE%E0%A4%AF_(%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%B0%E0%A4%BE%E0%A4%82%E0%A4%95)" title="पाय (स्थिरांक) – Marathi" lang="mr" hreflang="mr" data-title="पाय (स्थिरांक)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A8%D8%A7%D9%89_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="باى (رياضيات) – Egyptian Arabic" lang="arz" hreflang="arz" data-title="باى (رياضيات)" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pi" title="Pi – Malay" lang="ms" hreflang="ms" data-title="Pi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Pi" title="Pi – Minangkabau" lang="min" hreflang="min" data-title="Pi" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target">Minangkabau</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/I%C3%A8ng-ci%C5%AD-l%E1%B9%B3%CC%86k" title="Ièng-ciŭ-lṳ̆k – Mindong" lang="cdo" hreflang="cdo" data-title="Ièng-ciŭ-lṳ̆k" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Mongolian" lang="mn" hreflang="mn" data-title="Пи" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%95%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA_(%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC)" title="ပိုင် (သင်္ချာ) – Burmese" lang="my" hreflang="my" data-title="ပိုင် (သင်္ချာ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Pi" title="Pi – Fijian" lang="fj" hreflang="fj" data-title="Pi" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Pi (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/%E0%A4%AA%E0%A4%BE%E0%A4%88" title="पाई – Nepali" lang="ne" hreflang="ne" data-title="पाई" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%87" title="पाइ – Newari" lang="new" hreflang="new" data-title="पाइ" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%86%E5%91%A8%E7%8E%87" title="円周率 – Japanese" lang="ja" hreflang="ja" data-title="円周率" 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 badge-Q70894304 mw-list-item" title=""><a href="https://frr.wikipedia.org/wiki/Pi" title="Pi – Northern Frisian" lang="frr" hreflang="frr" data-title="Pi" 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/Pi" title="Pi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Pi" 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/Pi" title="Pi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Pi" 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/Pi" title="Pi – Occitan" lang="oc" hreflang="oc" data-title="Pi" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AA%E0%AC%BE%E0%AC%87" title="ପାଇ – Odia" lang="or" hreflang="or" data-title="ପାଇ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Pi" title="Pi – Uzbek" lang="uz" hreflang="uz" data-title="Pi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A8%BE%E0%A8%88" title="ਪਾਈ – Punjabi" lang="pa" hreflang="pa" data-title="ਪਾਈ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Kreiszahl" title="Kreiszahl – Palatine German" lang="pfl" hreflang="pfl" data-title="Kreiszahl" data-language-autonym="Pälzisch" data-language-local-name="Palatine German" class="interlanguage-link-target">Pälzisch</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%BE%D8%A7%D8%A6%DB%8C" title="پائی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="پائی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%BE%D8%A7%DB%8C_(_%D9%81%D8%B2%D9%8A%DA%A9_)" title="پای ( فزيک ) – Pashto" lang="ps" hreflang="ps" data-title="پای ( فزيک )" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Pi" title="Pi – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Pi" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Pi_(nombe)" title="Pi (nombe) – Picard" lang="pcd" hreflang="pcd" data-title="Pi (nombe)" data-language-autonym="Picard" data-language-local-name="Picard" class="interlanguage-link-target">Picard</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_%C3%ABd_Ludolph" title="Nùmer ëd Ludolph – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer ëd Ludolph" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krinktall" title="Krinktall – Low German" lang="nds" hreflang="nds" data-title="Krinktall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pi" title="Pi – Polish" lang="pl" hreflang="pl" data-title="Pi" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://pt.wikipedia.org/wiki/Pi" title="Pi – Portuguese" lang="pt" hreflang="pt" data-title="Pi" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ksh mw-list-item"><a href="https://ksh.wikipedia.org/wiki/Pi_(Kr%C3%A4j%C3%9Fzal)" title="Pi (Kräjßzal) – Colognian" lang="ksh" hreflang="ksh" data-title="Pi (Kräjßzal)" data-language-autonym="Ripoarisch" data-language-local-name="Colognian" class="interlanguage-link-target">Ripoarisch</a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ro.wikipedia.org/wiki/Pi" title="Pi – Romanian" lang="ro" hreflang="ro" data-title="Pi" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chiqaluwa" title="Chiqaluwa – Quechua" lang="qu" hreflang="qu" data-title="Chiqaluwa" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A7%D1%96%D1%81%D0%BB%D0%BE_%D0%BF%D1%96" title="Чісло пі – Rusyn" lang="rue" hreflang="rue" data-title="Чісло пі" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B8_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="Пи (число) – Russian" lang="ru" hreflang="ru" data-title="Пи (число)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Yakut" lang="sah" hreflang="sah" data-title="Пи" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्या – Sanskrit" lang="sa" hreflang="sa" data-title="प्या" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AF%E1%B1%9F%E1%B1%AD" title="ᱯᱟᱭ – Santali" lang="sat" hreflang="sat" data-title="ᱯᱟᱭ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Pi_grecu" title="Pi grecu – Sardinian" lang="sc" hreflang="sc" data-title="Pi grecu" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target">Sardu</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Pi" title="Pi – Scots" lang="sco" hreflang="sco" data-title="Pi" 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_pi" title="Numri pi – Albanian" lang="sq" hreflang="sq" data-title="Numri pi" 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/Pi_grecu" title="Pi grecu – Sicilian" lang="scn" hreflang="scn" data-title="Pi grecu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B6%BA%E0%B7%92_(%E0%B6%85%E0%B6%82%E0%B6%9A%E0%B6%BA)" title="පයි (අංකය) – Sinhala" lang="si" hreflang="si" data-title="පයි (අංකය)" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Pi" title="Pi – Simple English" lang="en-simple" hreflang="en-simple" data-title="Pi" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Ludolfovo_%C4%8D%C3%ADslo" title="Ludolfovo číslo – Slovak" lang="sk" hreflang="sk" data-title="Ludolfovo 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/Pi" title="Pi – Slovenian" lang="sl" hreflang="sl" data-title="Pi" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Pi" title="Pi – Silesian" lang="szl" hreflang="szl" data-title="Pi" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Summad_(Pi)" title="Summad (Pi) – Somali" lang="so" hreflang="so" data-title="Summad (Pi)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%D8%A7%DB%8C" title="پای – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پای" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B8" title="Пи – Serbian" lang="sr" hreflang="sr" data-title="Пи" 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/Pi" title="Pi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Pi" 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/Pii_(vakio)" title="Pii (vakio) – Finnish" lang="fi" hreflang="fi" data-title="Pii (vakio)" 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/Pi" title="Pi – Swedish" lang="sv" hreflang="sv" data-title="Pi" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pi" title="Pi – Tagalog" lang="tl" hreflang="tl" data-title="Pi" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%88_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AE%BF%E0%AE%B2%E0%AE%BF)" title="பை (கணித மாறிலி) – Tamil" lang="ta" hreflang="ta" data-title="பை (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/Pi" title="Pi – Tachelhit" lang="shi" hreflang="shi" data-title="Pi" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Pi" title="Pi – Kabyle" lang="kab" hreflang="kab" data-title="Pi" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-roa-tara mw-list-item"><a href="https://roa-tara.wikipedia.org/wiki/Pi_greche" title="Pi greche – Tarantino" lang="nap-x-tara" hreflang="nap-x-tara" data-title="Pi greche" data-language-autonym="Tarandíne" data-language-local-name="Tarantino" class="interlanguage-link-target">Tarandíne</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9F%D0%B8_%D1%81%D0%B0%D0%BD%D1%8B" title="Пи саны – Tatar" lang="tt" hreflang="tt" data-title="Пи саны" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B1%88" title="పై – Telugu" lang="te" hreflang="te" data-title="పై" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B2%E0%B8%A2_(%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%84%E0%B8%87%E0%B8%95%E0%B8%B1%E0%B8%A7)" title="พาย (ค่าคงตัว) – Thai" lang="th" hreflang="th" data-title="พาย (ค่าคงตัว)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9F%D3%A3_(%D0%B0%D0%B4%D0%B0%D0%B4)" title="Пӣ (адад) – Tajik" lang="tg" hreflang="tg" data-title="Пӣ (адад)" 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/Pi_say%C4%B1s%C4%B1" title="Pi sayısı – Turkish" lang="tr" hreflang="tr" data-title="Pi 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/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BF%D1%96" title="Число пі – Ukrainian" lang="uk" hreflang="uk" data-title="Число пі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%BE%D8%A7%D8%A6%DB%8C" title="پائی – Urdu" lang="ur" hreflang="ur" data-title="پائی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Pi_greco" title="Pi greco – Venetian" lang="vec" hreflang="vec" data-title="Pi greco" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Pi_(lugu)" title="Pi (lugu) – Veps" lang="vep" hreflang="vep" data-title="Pi (lugu)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://vi.wikipedia.org/wiki/Pi" title="Pi – Vietnamese" lang="vi" hreflang="vi" data-title="Pi" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Pii" title="Pii – Võro" lang="vro" hreflang="vro" data-title="Pii" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="圓周率" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pi" title="Pi – Waray" lang="war" hreflang="war" data-title="Pi" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 – Wu" lang="wuu" hreflang="wuu" data-title="圓周率" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%99" title="פי – Yiddish" lang="yi" hreflang="yi" data-title="פי" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Pi" title="Pi – Yoruba" lang="yo" hreflang="yo" data-title="Pi" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 – Cantonese" lang="yue" hreflang="yue" data-title="圓周率" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amar_Pi" title="Amar Pi – Dimli" lang="diq" hreflang="diq" data-title="Amar Pi" data-language-autonym="Zazaki" data-language-local-name="Dimli" class="interlanguage-link-target">Zazaki</a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Pi_(wiskunde)" title="Pi (wiskunde) – Zeelandic" lang="zea" hreflang="zea" data-title="Pi (wiskunde)" data-language-autonym="Zeêuws" data-language-local-name="Zeelandic" class="interlanguage-link-target">Zeêuws</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Pi" title="Pi – Samogitian" lang="sgs" hreflang="sgs" data-title="Pi" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://zh.wikipedia.org/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率 – Chinese" lang="zh" hreflang="zh" data-title="圓周率" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a 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<div class="mw-indicators"> <div id="mw-indicator-featured-star" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Featured_articles*" title="This is a featured article. Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> <div id="mw-indicator-pp-default" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected."><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 1.5x" data-file-width="512" data-file-height="512" /></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">Number, approximately 3.14</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">This article is about the mathematical constant. For the Greek letter, see <a href="/wiki/Pi_(letter)" title="Pi (letter)">Pi (letter)</a>. For other uses, see <a href="/wiki/Pi_(disambiguation)" class="mw-disambig" title="Pi (disambiguation)">Pi (disambiguation)</a> and <a href="/wiki/PI" class="mw-disambig" title="PI">PI</a>.</div> <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:": 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .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><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Pi" title="Category:Pi">a series of articles</a> on the</td></tr><tr><th class="sidebar-title-with-pretitle">mathematical constant <a class="mw-selflink selflink">π</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Pi-unrolled-720.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/220px-Pi-unrolled-720.gif" decoding="async" width="220" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/330px-Pi-unrolled-720.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/440px-Pi-unrolled-720.gif 2x" data-file-width="720" data-file-height="228" /></a></td></tr><tr><td class="sidebar-content"> 3.1415926535897932384626433...</td> </tr><tr><th class="sidebar-heading"> Uses</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area of a circle</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/List_of_formulae_involving_%CF%80" title="List of formulae involving π">Use in other formulae</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Properties</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">Irrationality</a></li> <li><a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Transcendence</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Value</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Less than 22/7</a></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations</a> <ul><li><a href="/wiki/Mil%C3%BC" title="Milü">Milü</a></li></ul></li> <li><a href="/wiki/Madhava%27s_correction_term" title="Madhava's correction term">Madhava's correction term</a></li> <li><a href="/wiki/Piphilology" title="Piphilology">Memorization</a></li></ul></td> </tr><tr><th class="sidebar-heading"> People</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Method_of_exhaustion#Archimedes" title="Method of exhaustion">Archimedes</a></li> <li><a href="/wiki/Liu_Hui%27s_%CF%80_algorithm" title="Liu Hui's π algorithm">Liu Hui</a></li> <li><a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava</a></li> <li><a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a></li> <li><a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Ludolph van Ceulen</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a></li> <li><a href="/wiki/Seki_Takakazu#Calculation_of_Pi" title="Seki Takakazu">Seki Takakazu</a></li> <li><a href="/wiki/Takebe_Kenko#Legacy" class="mw-redirect" title="Takebe Kenko"> Takebe Kenko</a></li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a></li> <li><a href="/wiki/John_Machin" title="John Machin">John Machin</a></li> <li><a href="/wiki/William_Shanks" title="William Shanks">William Shanks</a></li> <li><a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a></li> <li><a href="/wiki/John_Wrench" title="John Wrench">John Wrench</a></li> <li><a href="/wiki/Chudnovsky_brothers" title="Chudnovsky brothers">Chudnovsky brothers</a></li> <li><a href="/wiki/Yasumasa_Kanada" title="Yasumasa Kanada">Yasumasa Kanada</a></li></ul></td> </tr><tr><th class="sidebar-heading"> History</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">Chronology</a></li> <li><a href="/wiki/A_History_of_Pi" title="A History of Pi">A History of Pi</a></li></ul></td> </tr><tr><th class="sidebar-heading"> In culture</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Indiana_pi_bill" title="Indiana pi bill">Indiana pi bill</a></li> <li><a href="/wiki/Pi_Day" title="Pi Day">Pi Day</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Related topics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li> <li><a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a></li> <li><a href="/wiki/Six_nines_in_pi" title="Six nines in pi">Six nines in π</a></li> <li><a href="/wiki/List_of_topics_related_to_%CF%80" title="List of topics related to π">Other topics related to π</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:Pi_box" title="Template:Pi box"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Pi_box" title="Template talk:Pi box"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Pi_box" title="Special:EditPage/Template:Pi box"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> The number π (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/paɪ/</a> <span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/5e\/LL-Q1860_%28eng%29-Flame%2C_not_lame-Pi.wav\/LL-Q1860_%28eng%29-Flame%2C_not_lame-Pi.wav.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"LL-Q1860 (eng)-Flame, not lame-Pi.wav"},"classes":["noexcerpt","ext-phonos-PhonosButton","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5e/LL-Q1860_%28eng%29-Flame%2C_not_lame-Pi.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-Pi.wav.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"></a><a href="/wiki/File:LL-Q1860_(eng)-Flame,_not_lame-Pi.wav" title="File:LL-Q1860 (eng)-Flame, not lame-Pi.wav">ⓘ</a>; spelled out as pi) is a <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a>, approximately equal to 3.14159, that is the <a href="/wiki/Ratio" title="Ratio">ratio</a> of a <a href="/wiki/Circle" title="Circle">circle</a>'s <a href="/wiki/Circumference" title="Circumference">circumference</a> to its <a href="/wiki/Diameter" title="Diameter">diameter</a>. It appears in many formulae across <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Physics" title="Physics">physics</a>, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the <a href="/wiki/Arc_length" title="Arc length">length of a curve</a>. The number π is an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {22}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>22</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {22}{7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb5d075d03ff7edd66274d09cb1e2dd0aee7bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {22}{7}}}" /> are commonly <a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">used to approximate it</a>. Consequently, its <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a> never ends, nor <a href="/wiki/Repeating_decimal" title="Repeating decimal">enters a permanently repeating pattern</a>. It is a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>, meaning that it cannot be a solution of an <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equation</a> involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a> with a <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass and straightedge</a>. The decimal digits of π appear to be <a href="/wiki/Random_sequence" title="Random sequence">randomly distributed</a>,<a href="#cite_note-1">[a]</a> but no proof of this <a href="/wiki/Conjecture" title="Conjecture">conjecture</a> has been found. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptians</a> and <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonians</a>, required fairly accurate approximations of π for practical computations. Around 250 BC, the <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematician</a> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematicians</a> approximated π to seven digits, while <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematicians</a> made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>, was discovered a millennium later.<a href="#cite_note-FOOTNOTEAndrewsAskeyRoy199959-2">[1]</a><a href="#cite_note-3">[2]</a> The earliest known use of the Greek letter <a href="/wiki/Pi_(letter)" title="Pi (letter)">π</a> to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician <a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> in 1706.<a href="#cite_note-jones-4">[3]</a> The invention of <a href="/wiki/Calculus" title="Calculus">calculus</a> soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and <a href="/wiki/Computer_science" title="Computer science">computer scientists</a> have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits.<a href="#cite_note-5">[4]</a><a href="#cite_note-6">[5]</a> These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records.<a href="#cite_note-FOOTNOTEArndtHaenel200617-7">[6]</a><a href="#cite_note-8">[7]</a> The extensive computations involved have also been used to test <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a> as well as stress testing consumer computer hardware. Because it relates to a circle, π is found in many formulae in <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>, <a href="/wiki/Fractal" title="Fractal">fractals</a>, <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, and <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>. It also appears in areas having little to do with geometry, such as <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, and in modern <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> can be defined without any reference to geometry. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Fundamentals">Fundamentals</h2></div> <div class="mw-heading mw-heading3"><h3 id="Name">Name</h3></div> The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase <a href="/wiki/Pi_(letter)" title="Pi (letter)">Greek letter π</a>, sometimes spelled out as pi.<a href="#cite_note-firstPi-9">[8]</a> In English, π is <a href="/wiki/English_pronunciation_of_Greek_letters" class="mw-redirect" title="English pronunciation of Greek letters">pronounced as "pie"</a> (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/paɪ/</a> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key">PY</a>).<a href="#cite_note-10">[9]</a> In mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart Π, which denotes a <a href="/wiki/Multiplication#Product_of_a_sequence" title="Multiplication">product of a sequence</a>, analogous to how Σ denotes <a href="/wiki/Summation" title="Summation">summation</a>. The choice of the symbol π is discussed in the section <a href="#Adoption_of_the_symbol_π">Adoption of the symbol π</a>. <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pi_eq_C_over_d.svg" class="mw-file-description"><img alt="A diagram of a circle, with the width labelled as diameter, and the perimeter labelled as circumference" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Pi_eq_C_over_d.svg/220px-Pi_eq_C_over_d.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Pi_eq_C_over_d.svg/330px-Pi_eq_C_over_d.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Pi_eq_C_over_d.svg/440px-Pi_eq_C_over_d.svg.png 2x" data-file-width="512" data-file-height="510" /></a><figcaption>The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.</figcaption></figure> π is commonly defined as the <a href="/wiki/Ratio" title="Ratio">ratio</a> of a <a href="/wiki/Circle" title="Circle">circle</a>'s <a href="/wiki/Circumference" title="Circumference">circumference</a> C to its <a href="/wiki/Diameter" title="Diameter">diameter</a> d:<a href="#cite_note-FOOTNOTEArndtHaenel20068-11">[10]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {C}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {C}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ffc6970dda2e60597854f14e2ac1e13a25a5cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.033ex; height:5.509ex;" alt="{\displaystyle \pi ={\frac {C}{d}}}" /> The ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {C}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {C}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ebfc4cd5de67be49da2841f490d1d55053a1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.085ex; height:3.843ex;" alt="{\textstyle {\frac {C}{d}}}" /> is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {C}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {C}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ebfc4cd5de67be49da2841f490d1d55053a1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.085ex; height:3.843ex;" alt="{\textstyle {\frac {C}{d}}}" />. This definition of π implicitly makes use of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">flat (Euclidean) geometry</a>; although the notion of a circle can be extended to any <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">curve (non-Euclidean) geometry</a>, these new circles will no longer satisfy the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi ={\frac {C}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi ={\frac {C}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a38da24b191fd9293cb427b50587c924cd97824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.516ex; height:3.843ex;" alt="{\textstyle \pi ={\frac {C}{d}}}" />.<a href="#cite_note-FOOTNOTEArndtHaenel20068-11">[10]</a> Here, the circumference of a circle is the <a href="/wiki/Arc_length" title="Arc length">arc length</a> around the <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> of the circle, a quantity which can be formally defined independently of geometry using <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>—a concept in <a href="/wiki/Calculus" title="Calculus">calculus</a>.<a href="#cite_note-12">[11]</a> For example, one may directly compute the arc length of the top half of the unit circle, given in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x^{2}+y^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1207955c0a974b8d1849a63adbfb457d9fd4679c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:2.843ex;" alt="{\textstyle x^{2}+y^{2}=1}" />, as the <a href="/wiki/Integral" title="Integral">integral</a>:<a href="#cite_note-FOOTNOTERemmert2012129-13">[12]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50725fce8b20701f737a6ecc57aff4b8969107c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.636ex; height:7.009ex;" alt="{\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}" /> An integral such as this was proposed as a definition of π by <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>, who defined it directly as an integral in 1841.<a href="#cite_note-15">[b]</a> Integration is no longer commonly used in a first analytical definition because, as <a href="#CITEREFRemmert2012">Remmert 2012</a> explains, <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer<a href="#cite_note-16">[14]</a> and popularized by <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a>,<a href="#cite_note-17">[15]</a> is the following: π is twice the smallest positive number at which the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> function equals 0.<a href="#cite_note-FOOTNOTEArndtHaenel20068-11">[10]</a><a href="#cite_note-FOOTNOTERemmert2012129-13">[12]</a><a href="#cite_note-Rudin_1976-18">[16]</a> π is also the smallest positive number at which the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a <a href="/wiki/Power_series" title="Power series">power series</a>,<a href="#cite_note-19">[17]</a> or as the solution of a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>.<a href="#cite_note-Rudin_1976-18">[16]</a> In a similar spirit, π can be defined using properties of the <a href="/wiki/Complex_exponential" class="mw-redirect" title="Complex exponential">complex exponential</a>, exp z, of a <a href="/wiki/Complex_number" title="Complex number">complex</a> variable z. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\dots ,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>,</mo> <mn>4</mn> <mi>π</mi> <mi>i</mi> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>i</mi> <mo>∣</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\dots ,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9411506ae82c49d1b868a47ce5dea04adf234cba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.66ex; height:2.843ex;" alt="{\displaystyle \{\dots ,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}}" /> and there is a unique positive real number π with this property.<a href="#cite_note-FOOTNOTERemmert2012129-13">[12]</a><a href="#cite_note-20">[18]</a> A variation on the same idea, making use of sophisticated mathematical concepts of <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Algebra" title="Algebra">algebra</a>, is the following theorem:<a href="#cite_note-21">[19]</a> there is a unique (<a href="/wiki/Up_to" title="Up to">up to</a> <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>) <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> from the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> R/Z of real numbers under addition <a href="/wiki/Quotient_group" title="Quotient group">modulo</a> integers (the <a href="/wiki/Circle_group" title="Circle group">circle group</a>), onto the multiplicative group of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> of <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> one. The number π is then defined as half the magnitude of the derivative of this homomorphism.<a href="#cite_note-Nicolas_Bourbaki-22">[20]</a> <div class="mw-heading mw-heading3"><h3 id="Irrationality_and_normality">Irrationality and normality</h3></div> π is an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, meaning that it cannot be written as the <a href="/wiki/Rational_number" title="Rational number">ratio of two integers</a>. Fractions such as <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>⁠22/7⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠355/113⁠ are commonly used to approximate π, but no <a href="/wiki/Common_fraction" class="mw-redirect" title="Common fraction">common fraction</a> (ratio of whole numbers) can be its exact value.<a href="#cite_note-FOOTNOTEArndtHaenel20065-23">[21]</a> Because π is irrational, it has an infinite number of digits in its <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a>, and does not settle into an infinitely <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating pattern</a> of digits. There are several <a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">proofs that π is irrational</a>; they are generally <a href="/wiki/Proofs_by_contradiction" class="mw-redirect" title="Proofs by contradiction">proofs by contradiction</a> and require calculus. The degree to which π can be approximated by <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (called the <a href="/wiki/Irrationality_measure" title="Irrationality measure">irrationality measure</a>) is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of e but smaller than the measure of <a href="/wiki/Liouville_number" title="Liouville number">Liouville numbers</a>.<a href="#cite_note-24">[22]</a> The digits of π have no apparent pattern and have passed tests for <a href="/wiki/Statistical_randomness" title="Statistical randomness">statistical randomness</a>, including tests for <a href="/wiki/Normal_number" title="Normal number">normality</a>; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that π is <a href="/wiki/Normal_number" title="Normal number">normal</a> has not been proven or disproven.<a href="#cite_note-FOOTNOTEArndtHaenel200622–23-25">[23]</a> Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. <a href="/wiki/Yasumasa_Kanada" title="Yasumasa Kanada">Yasumasa Kanada</a> has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to <a href="/wiki/Statistical_significance_test" class="mw-redirect" title="Statistical significance test">statistical significance tests</a>, and no evidence of a pattern was found.<a href="#cite_note-FOOTNOTEArndtHaenel200622,_28–30-26">[24]</a> Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the <a href="/wiki/Infinite_monkey_theorem" title="Infinite monkey theorem">infinite monkey theorem</a>. Thus, because the sequence of π's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a <a href="/wiki/Six_nines_in_pi" title="Six nines in pi">sequence of six consecutive 9s</a> that begins at the 762nd decimal place of the decimal representation of π.<a href="#cite_note-FOOTNOTEArndtHaenel20063-27">[25]</a> This is also called the "Feynman point" in <a href="/wiki/Mathematical_folklore" title="Mathematical folklore">mathematical folklore</a>, after <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>, although no connection to Feynman is known. <div class="mw-heading mw-heading3"><h3 id="Transcendence">Transcendence</h3></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/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a></div><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Squaring_the_circle.svg" class="mw-file-description"><img alt="A diagram of a square and circle, both with identical area; the length of the side of the square is the square root of pi" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/220px-Squaring_the_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/330px-Squaring_the_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/440px-Squaring_the_circle.svg.png 2x" data-file-width="281" data-file-height="281" /></a><figcaption>Because π is a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>, <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a> is not possible in a finite number of steps using the classical tools of <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass and straightedge</a>.</figcaption></figure> In addition to being irrational, π is also a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>, which means that it is not the <a href="/wiki/Solution_(equation)" class="mw-redirect" title="Solution (equation)">solution</a> of any non-constant <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>120</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb75c4d7ee0e26b7281831ffb74ea6d2496b1b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.181ex; height:4.176ex;" alt="{\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}" />.<a href="#cite_note-FOOTNOTEArndtHaenel20066-28">[26]</a><a href="#cite_note-29">[c]</a> This follows from the so-called <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem#Transcendence_of_e_and_π" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a>, which also establishes the transcendence of <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">the constant e</a>. The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or <a href="/wiki/Nth_root" title="Nth root">n-th roots</a> (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{31}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>31</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{31}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb1cd8b2742403364f3adca4fc8b8ca4a21e7f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{31}}}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7409b0ddbc1f90280265e7bc95dd20626ebf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {10}}}" />). Second, since no transcendental number can be <a href="/wiki/Constructible_number" title="Constructible number">constructed</a> with <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass and straightedge</a>, it is not possible to "<a href="/wiki/Squaring_the_circle" title="Squaring the circle">square the circle</a>". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.<a href="#cite_note-30">[27]</a> Squaring a circle was one of the important geometry problems of the <a href="/wiki/Classical_antiquity" title="Classical antiquity">classical antiquity</a>.<a href="#cite_note-31">[28]</a> Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.<a href="#cite_note-32">[29]</a><a href="#cite_note-33">[30]</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 π and e are <a href="/wiki/Algebraic_independence" title="Algebraic independence">algebraically independent</a> ("relatively transcendental"). This would be resolved by <a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a><a href="#cite_note-34">[31]</a><a href="#cite_note-35">[32]</a> – a currently unproven generalization of the Lindemann–Weierstrass theorem.<a href="#cite_note-36">[33]</a> <div class="mw-heading mw-heading3"><h3 id="Continued_fractions">Continued fractions</h3></div> As an irrational number, π cannot be represented as a <a href="/wiki/Common_fraction" class="mw-redirect" title="Common fraction">common fraction</a>. But every number, including π, can be represented by an infinite series of nested fractions, called a <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fraction</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>292</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e294255007892e9424e6d3f49183cad6b133bbcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.005ex; width:48.772ex; height:30.009ex;" alt="{\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}" /> Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠22/7⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠333/106⁠, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠355/113⁠. These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.<a href="#cite_note-Eymard_1999_78-37">[34]</a> Because π is transcendental, it is by definition not <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic</a> and so cannot be a <a href="/wiki/Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrational</a>. Therefore, π cannot have a <a href="/wiki/Periodic_continued_fraction" title="Periodic continued fraction">periodic continued fraction</a>. Although the simple continued fraction for π (with numerators all 1, shown above) also does not exhibit any other obvious pattern,<a href="#cite_note-FOOTNOTEArndtHaenel200633-38">[35]</a><a href="#cite_note-mollin-39">[36]</a> several non-simple <a href="/wiki/Continued_fraction" title="Continued fraction">continued fractions</a> do, such as:<a href="#cite_note-40">[37]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi &=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></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></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi &=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c2f88e03c466938a66d99b268ff5a5ea756dc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.94ex; margin-bottom: -0.231ex; width:82.386ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\pi &=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}\end{aligned}}}" /> The middle of these is due to the mid-17th century mathematician <a href="/wiki/William_Brouncker,_2nd_Viscount_Brouncker" title="William Brouncker, 2nd Viscount Brouncker">William Brouncker</a>, see <a href="/wiki/William_Brouncker,_2nd_Viscount_Brouncker#Brouncker's_formula" title="William Brouncker, 2nd Viscount Brouncker">§ Brouncker's formula</a>. <div class="mw-heading mw-heading3"><h3 id="Approximate_value_and_digits">Approximate value and digits</h3></div> Some <a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">approximations of pi</a> include: <ul><li>Integers: 3</li> <li>Fractions: Approximate fractions include (in order of increasing accuracy) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠22/7⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠333/106⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠355/113⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠52163/16604⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠103993/33102⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠104348/33215⁠, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠245850922/78256779⁠.<a href="#cite_note-Eymard_1999_78-37">[34]</a> (List is selected terms from <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A063674" class="extiw" title="oeis:A063674">A063674</a> and <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A063673" class="extiw" title="oeis:A063673">A063673</a>.)</li> <li>Digits: The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510...<a href="#cite_note-FOOTNOTEArndtHaenel2006240-41">[38]</a> (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000796" class="extiw" title="oeis:A000796">A000796</a>)</li></ul> Digits in other number systems <ul><li>The first 48 <a href="/wiki/Binary_number#Representing_real_numbers" title="Binary number">binary</a> (<a href="/wiki/Radix" title="Radix">base</a> 2) digits (called <a href="/wiki/Bit" title="Bit">bits</a>) are 11.001001000011111101101010100010001000010110100011... (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A004601" class="extiw" title="oeis:A004601">A004601</a>)</li> <li>The first 36 digits in <a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">ternary</a> (base 3) are 10.010211012222010211002111110221222220... (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A004602" class="extiw" title="oeis:A004602">A004602</a>)</li> <li>The first 20 digits in <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> (base 16) are 3.243F6A8885A308D31319...<a href="#cite_note-FOOTNOTEArndtHaenel2006242-42">[39]</a> (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A062964" class="extiw" title="oeis:A062964">A062964</a>)</li> <li>The first five <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> (base 60) digits are 3;8,29,44,0,47<a href="#cite_note-43">[40]</a> (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A060707" class="extiw" title="oeis:A060707">A060707</a>)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers_and_Euler's_identity">Complex numbers and Euler's identity</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Euler%27s_formula.svg" class="mw-file-description"><img alt="A diagram of a unit circle centred at the origin in the complex plane, including a ray from the centre of the circle to its edge, with the triangle legs labelled with sine and cosine functions." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/220px-Euler%27s_formula.svg.png" decoding="async" width="220" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/330px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/440px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a><figcaption>The association between imaginary powers of the number e and <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> centred at the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> given by <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></figcaption></figure> Any <a href="/wiki/Complex_number" title="Complex number">complex number</a>, say z, can be expressed using a pair of <a href="/wiki/Real_number" title="Real number">real numbers</a>. In the <a href="/wiki/Polar_coordinate_system#Complex_numbers" title="Polar coordinate system">polar coordinate system</a>, one number (<a href="/wiki/Radius" title="Radius">radius</a> or r) is used to represent z's distance from the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, and the other (angle or φ) the counter-clockwise <a href="/wiki/Rotation" title="Rotation">rotation</a> from the positive real line:<a href="#cite_note-FOOTNOTEAbramson2014[httpsopenstaxorgbooksprecalculuspages8-5-polar-form-of-complex-numbers_Section_8.5:_Polar_form_of_complex_numbers]-44">[41]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e93279bb31b398ca08d16fb9d764fd1214174ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.182ex; height:2.843ex;" alt="{\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ),}" /> where i is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}" />. The frequent appearance of π in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> can be related to the behaviour of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> of a complex variable, described by <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>:<a href="#cite_note-EF-45">[42]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b85e2656dda51f900210b347fb30b65d3a26f11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.515ex; height:3.176ex;" alt="{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,}" /> where <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">the constant e</a> is the base of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. This formula establishes a correspondence between imaginary powers of e and points on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> centred at the origin of the complex plane. Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f181a336ae10f53a432232ff83696fbdeb4e622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.951ex; height:2.176ex;" alt="{\displaystyle \varphi =\pi }" /> in Euler's formula results in <a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a>, celebrated in mathematics due to it containing five important mathematical constants:<a href="#cite_note-EF-45">[42]</a><a href="#cite_note-46">[43]</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> </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/e76caf050d8bc37cd2350c40517face26de5ecb7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.736ex; height:2.843ex;" alt="{\displaystyle e^{i\pi }+1=0.}" /> There are n different <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> z satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fb31edf665701c275c44a5be8b82a95509888d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.57ex; height:2.343ex;" alt="{\displaystyle z^{n}=1}" />, and these are called the "n-th <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a>"<a href="#cite_note-FOOTNOTEAndrewsAskeyRoy199914-47">[44]</a> and are given by the formula: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mspace width="2em"></mspace> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74d8393eb2cf4d7a3e5ef0dd9ab50a9fe58389f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.854ex; height:3.343ex;" alt="{\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></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/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</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/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">Chronology of computation of π</a></div> Surviving ancient approximations of π, before the second century AD, are accurate to one or two decimal places at best. The earliest written approximations are found in <a href="/wiki/Babylon" title="Babylon">Babylon</a> and Egypt, both within one percent of the true value. In Babylon, a <a href="/wiki/Clay_tablet" title="Clay tablet">clay tablet</a> dated 1900–1600 BC has a geometrical statement that, by implication, treats π as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠25/8⁠ = 3.125.<a href="#cite_note-FOOTNOTEArndtHaenel2006167-48">[45]</a> In Egypt, the <a href="/wiki/Rhind_Papyrus" class="mw-redirect" title="Rhind Papyrus">Rhind Papyrus</a>, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats π as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>9</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>3.16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e578ff1826354c928fbf5aeef8660fcaa0f388d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.897ex; height:4.009ex;" alt="{\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16}" />.<a href="#cite_note-mollin-39">[36]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006167-48">[45]</a> Although some <a href="/wiki/Pyramidology" title="Pyramidology">pyramidologists</a> have theorized that the <a href="/wiki/Great_Pyramid_of_Giza" title="Great Pyramid of Giza">Great Pyramid of Giza</a> was built with proportions related to π, this theory is not widely accepted by scholars.<a href="#cite_note-49">[46]</a> In the <a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a> of <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a>, dating to an oral tradition from the 1st or 2nd millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.<a href="#cite_note-50">[47]</a> <div class="mw-heading mw-heading3"><h3 id="Polygon_approximation_era">Polygon approximation era</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Domenico-Fetti_Archimedes_1620.jpg" class="mw-file-description"><img alt="A painting of a man studying" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/220px-Domenico-Fetti_Archimedes_1620.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/330px-Domenico-Fetti_Archimedes_1620.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Domenico-Fetti_Archimedes_1620.jpg/440px-Domenico-Fetti_Archimedes_1620.jpg 2x" data-file-width="1364" data-file-height="1818" /></a><figcaption><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> developed the polygonal approach to approximating π.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_pi.svg" class="mw-file-description"><img alt="diagram of a hexagon and pentagon circumscribed outside a circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/220px-Archimedes_pi.svg.png" decoding="async" width="220" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/330px-Archimedes_pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/440px-Archimedes_pi.svg.png 2x" data-file-width="750" data-file-height="250" /></a><figcaption>π can be estimated by computing the perimeters of circumscribed and inscribed polygons.</figcaption></figure> The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>, implementing the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>.<a href="#cite_note-FOOTNOTEArndtHaenel2006170-51">[48]</a> This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as Archimedes's constant.<a href="#cite_note-FOOTNOTEArndtHaenel2006175,_205-52">[49]</a> Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠223/71⁠ < π < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠22/7⁠ (that is, 3.1408 < π < 3.1429).<a href="#cite_note-life-of-pi-53">[50]</a> (Archimedes' upper bound of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠22/7⁠ may have led to a widespread popular belief that π is equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠22/7⁠.<a href="#cite_note-FOOTNOTEArndtHaenel2006171-54">[51]</a>) Around 150 AD, Greco-Roman scientist <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>, in his <a href="/wiki/Almagest" title="Almagest">Almagest</a>, gave a value for π of 3.1416, which he may have obtained from Archimedes or from <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a>.<a href="#cite_note-FOOTNOTEArndtHaenel2006176-55">[52]</a><a href="#cite_note-FOOTNOTEBoyerMerzbach1991168-56">[53]</a> Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.<a href="#cite_note-ArPI-57">[54]</a> In <a href="/wiki/Ancient_China" class="mw-redirect" title="Ancient China">ancient China</a>, values for π included 3.1547 (around 1 AD), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7409b0ddbc1f90280265e7bc95dd20626ebf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {10}}}" /> (100 AD, approximately 3.1623), and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠142/45⁠ (3rd century, approximately 3.1556).<a href="#cite_note-FOOTNOTEArndtHaenel2006176–177-58">[55]</a> Around 265 AD, the <a href="/wiki/Cao_Wei" title="Cao Wei">Cao Wei</a> mathematician <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> created a <a href="/wiki/Liu_Hui%27s_%CF%80_algorithm" title="Liu Hui's π algorithm">polygon-based iterative algorithm</a>, with which he constructed a 3,072-sided polygon to approximate π as 3.1416.<a href="#cite_note-autogenerated202-59">[56]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006177-60">[57]</a> Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.<a href="#cite_note-autogenerated202-59">[56]</a> Around 480 AD, <a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> calculated that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.1415926<\pi <3.1415927}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.1415926</mn> <mo><</mo> <mi>π</mi> <mo><</mo> <mn>3.1415927</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.1415926<\pi <3.1415927}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aca8712bcbd7836f7e1db9811a1e93ef71110aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.422ex; height:2.176ex;" alt="{\displaystyle 3.1415926<\pi <3.1415927}" /> and suggested the approximations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi \approx {\frac {355}{113}}=3.14159292035...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>355</mn> <mn>113</mn> </mfrac> </mrow> <mo>=</mo> <mn>3.14159292035...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi \approx {\frac {355}{113}}=3.14159292035...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06692ba6599484d7d1972c90781be9b2bdd86471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.368ex; height:3.676ex;" alt="{\textstyle \pi \approx {\frac {355}{113}}=3.14159292035...}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi \approx {\frac {22}{7}}=3.142857142857...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>22</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mn>3.142857142857...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi \approx {\frac {22}{7}}=3.142857142857...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/778369e629e3921a3abeddfa34954b1022abe1e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.708ex; height:3.676ex;" alt="{\textstyle \pi \approx {\frac {22}{7}}=3.142857142857...}" />, which he termed the <a href="/wiki/Mil%C3%BC" title="Milü">milü</a> ('close ratio') and yuelü ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of π for the next 800 years.<a href="#cite_note-FOOTNOTEArndtHaenel2006178-61">[58]</a> The Indian astronomer <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> used a value of 3.1416 in his <a href="/wiki/%C4%80ryabha%E1%B9%AD%C4%ABya" class="mw-redirect" title="Āryabhaṭīya">Āryabhaṭīya</a> (499 AD).<a href="#cite_note-FOOTNOTEArndtHaenel2006179-62">[59]</a> Around 1220, <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> computed 3.1418 using a polygonal method devised independently of Archimedes.<a href="#cite_note-FOOTNOTEArndtHaenel2006180-63">[60]</a> Italian author <a href="/wiki/Dante" class="mw-redirect" title="Dante">Dante</a> apparently employed the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>10</mn> </mfrac> </mrow> <mo>≈</mo> <mn>3.14142</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7acb14f07f1be2393396e80d85e48abd0da0539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.75ex; height:4.343ex;" alt="{\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}" />.<a href="#cite_note-FOOTNOTEArndtHaenel2006180-63">[60]</a> The Persian astronomer <a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> produced nine <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 3\times 2^{28}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>28</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 3\times 2^{28}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4f51cc91785a99c47ef48b6828cb4cf0020195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.042ex; height:2.676ex;" alt="{\textstyle 3\times 2^{28}}" /> sides,<a href="#cite_note-64">[61]</a><a href="#cite_note-65">[62]</a> which stood as the world record for about 180 years.<a href="#cite_note-FOOTNOTEArndtHaenel2006182-66">[63]</a> French mathematician <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> in 1579 achieved nine digits with a polygon of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 3\times 2^{17}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 3\times 2^{17}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c6b820448824185aa6348247f55cc2490cceb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.042ex; height:2.676ex;" alt="{\textstyle 3\times 2^{17}}" /> sides.<a href="#cite_note-FOOTNOTEArndtHaenel2006182-66">[63]</a> Flemish mathematician <a href="/wiki/Adriaan_van_Roomen" title="Adriaan van Roomen">Adriaan van Roomen</a> arrived at 15 decimal places in 1593.<a href="#cite_note-FOOTNOTEArndtHaenel2006182-66">[63]</a> In 1596, Dutch mathematician <a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Ludolph van Ceulen</a> reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).<a href="#cite_note-FOOTNOTEArndtHaenel2006182–183-67">[64]</a> Dutch scientist <a href="/wiki/Willebrord_Snellius" title="Willebrord Snellius">Willebrord Snellius</a> reached 34 digits in 1621,<a href="#cite_note-FOOTNOTEArndtHaenel2006183-68">[65]</a> and Austrian astronomer <a href="/wiki/Christoph_Grienberger" title="Christoph Grienberger">Christoph Grienberger</a> arrived at 38 digits in 1630 using 1040 sides.<a href="#cite_note-69">[66]</a> <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to <a href="/wiki/Richardson_extrapolation" title="Richardson extrapolation">Richardson extrapolation</a>.<a href="#cite_note-70">[67]</a><a href="#cite_note-71">[68]</a> <div class="mw-heading mw-heading3"><h3 id="Infinite_series">Infinite series</h3></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Comparison_pi_infinite_series.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Comparison_pi_infinite_series.svg/300px-Comparison_pi_infinite_series.svg.png" decoding="async" width="300" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Comparison_pi_infinite_series.svg/450px-Comparison_pi_infinite_series.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Comparison_pi_infinite_series.svg/600px-Comparison_pi_infinite_series.svg.png 2x" data-file-width="512" data-file-height="256" /></a><figcaption>Comparison of the convergence of several historical infinite series for π. Sn is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. <a href="/wiki/File:Comparison_pi_infinite_series.svg" title="File:Comparison pi infinite series.svg">(click for detail)</a> </figcaption></figure> The calculation of π was revolutionized by the development of <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a>. Infinite series allowed mathematicians to compute π with much greater precision than <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> and others who used geometrical techniques.<a href="#cite_note-Ais-72">[69]</a> Although infinite series were exploited for π most notably by European mathematicians such as <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, the approach also appeared in the <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school</a> sometime in the 14th or 15th century.<a href="#cite_note-Roypp-73">[70]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006185–186-74">[71]</a> Around 1500 AD, a written description of an infinite series that could be used to compute π was laid out in <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> verse in <a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a> by <a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a>.<a href="#cite_note-Roypp-73">[70]</a> The series are presented without proof, but proofs are presented in a later work, <a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a>, from around 1530 AD. Several infinite series are described, including series for sine (which Nilakantha attributes to <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a>), cosine, and arctangent which are now sometimes referred to as <a href="/wiki/Madhava_series" title="Madhava series">Madhava series</a>. The series for arctangent is sometimes called <a href="/wiki/Gregory%27s_series" class="mw-redirect" title="Gregory's series">Gregory's series</a> or the Gregory–Leibniz series.<a href="#cite_note-Roypp-73">[70]</a> Madhava used infinite series to estimate π to 11 digits around 1400.<a href="#cite_note-75">[72]</a> In 1593, <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> published what is now known as <a href="/wiki/Vi%C3%A8te%27s_formula" title="Viète's formula">Viète's formula</a>, an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a> (rather than an <a href="/wiki/Infinite_sum" class="mw-redirect" title="Infinite sum">infinite sum</a>, which is more typically used in π calculations):<a href="#cite_note-FOOTNOTEArndtHaenel2006187-76">[73]</a><a href="#cite_note-77">[74]</a><a href="#cite_note-78">[75]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msqrt> </mrow> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/471259d8063f909988b7457a3e2d9c62f73efb01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.518ex; height:9.009ex;" alt="{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }" /> In 1655, <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> published what is now known as <a href="/wiki/Wallis_product" title="Wallis product">Wallis product</a>, also an infinite product:<a href="#cite_note-FOOTNOTEArndtHaenel2006187-76">[73]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>1</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>7</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>9</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c54735394beda8b94805a2919f1f4c0e200d2b29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.221ex; height:5.343ex;" alt="{\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots }" /> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GodfreyKneller-IsaacNewton-1689.jpg" class="mw-file-description"><img alt="A formal portrait of a man, with long hair" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/250px-GodfreyKneller-IsaacNewton-1689.jpg" decoding="async" width="170" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/330px-GodfreyKneller-IsaacNewton-1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/500px-GodfreyKneller-IsaacNewton-1689.jpg 2x" data-file-width="1364" data-file-height="1916" /></a><figcaption><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> used <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".<a href="#cite_note-Newton-79">[76]</a></figcaption></figure> In the 1660s, the English scientist <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and German mathematician <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> discovered <a href="/wiki/Calculus" title="Calculus">calculus</a>, which led to the development of many infinite series for approximating π. Newton himself used an arcsine series to compute a 15-digit approximation of π in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."<a href="#cite_note-Newton-79">[76]</a> In 1671, <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a>, and independently, Leibniz in 1673, discovered the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion for <a href="/wiki/Arctangent" class="mw-redirect" title="Arctangent">arctangent</a>:<a href="#cite_note-Roypp-73">[70]</a><a href="#cite_note-80">[77]</a><a href="#cite_note-LS-81">[78]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0e80b93f9981701b98f8e121e27c4c50955a3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.155ex; height:5.843ex;" alt="{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }" /> This series, sometimes called the <a href="/wiki/Gregory%27s_series" class="mw-redirect" title="Gregory's series">Gregory–Leibniz series</a>, equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc79dd123359dbfd89513870537035bcbf0dd6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\textstyle {\frac {\pi }{4}}}" /> when evaluated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" />.<a href="#cite_note-LS-81">[78]</a> But for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" />, <a href="/wiki/Leibniz_formula_for_%CF%80#Convergence" title="Leibniz formula for π">it converges impractically slowly</a> (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.<a href="#cite_note-82">[79]</a> In 1699, English mathematician <a href="/wiki/Abraham_Sharp" title="Abraham Sharp">Abraham Sharp</a> used the Gregory–Leibniz series for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z={\frac {1}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z={\frac {1}{\sqrt {3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae2125e0304d50f14a14f9559d3611557bbceaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.214ex; height:4.176ex;" alt="{\textstyle z={\frac {1}{\sqrt {3}}}}" /> to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.<a href="#cite_note-FOOTNOTEArndtHaenel2006189-83">[80]</a> In 1706, <a href="/wiki/John_Machin" title="John Machin">John Machin</a> used the Gregory–Leibniz series to produce an algorithm that converged much faster:<a href="#cite_note-jones-4">[3]</a><a href="#cite_note-tweddle-84">[81]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006192–193-85">[82]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>239</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e27dbc5ad6b84571aaaae627e1d0080145e6a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.332ex; height:5.176ex;" alt="{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}" /> Machin reached 100 digits of π with this formula.<a href="#cite_note-A72n4-86">[83]</a> Other mathematicians created variants, now known as <a href="/wiki/Machin-like_formula" title="Machin-like formula">Machin-like formulae</a>, that were used to set several successive records for calculating digits of π.<a href="#cite_note-87">[84]</a><a href="#cite_note-A72n4-86">[83]</a> Isaac Newton <a href="/wiki/Series_acceleration" title="Series acceleration">accelerated the convergence</a> of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):<a href="#cite_note-88">[85]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6282b6fd3849a778a8897b3e40e25448e427a0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.76ex; height:6.509ex;" alt="{\displaystyle \arctan x={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots }" /></dd></dl> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>5</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>79</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202bbce8b648f8d1be219e9ea77aa0bc21230d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.307ex; height:3.676ex;" alt="{\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},}" /> with which he computed 20 digits of π in one hour.<a href="#cite_note-89">[86]</a> Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.<a href="#cite_note-FOOTNOTEArndtHaenel2006192–196,_205-90">[87]</a> In 1844, a record was set by <a href="/wiki/Zacharias_Dase" title="Zacharias Dase">Zacharias Dase</a>, who employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>.<a href="#cite_note-A194-91">[88]</a> In 1853, British mathematician <a href="/wiki/William_Shanks" title="William Shanks">William Shanks</a> calculated π to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.<a href="#cite_note-hayes-2014-92">[89]</a> <div class="mw-heading mw-heading4"><h4 id="Rate_of_convergence">Rate of convergence</h4></div> Some infinite series for π <a href="/wiki/Convergent_series" title="Convergent series">converge</a> faster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.<a href="#cite_note-Aconverge-93">[90]</a> A simple infinite series for π is the <a href="/wiki/Leibniz_formula_for_%CF%80" title="Leibniz formula for π">Gregory–Leibniz series</a>:<a href="#cite_note-FOOTNOTEArndtHaenel200669–72-94">[91]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>11</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>13</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e3959cd2d0ec735e7a6a1917df784842b76706" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.352ex; height:5.343ex;" alt="{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots }" /> As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and – with a sufficient number of terms – can get as close to π as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of π.<a href="#cite_note-95">[92]</a> An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:<a href="#cite_note-FOOTNOTEArndtHaenel2006Formula_16.10,_p._223-96">[93]</a><a href="#cite_note-97">[94]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>4</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>6</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>8</mn> <mo>×</mo> <mn>9</mn> <mo>×</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdafa8bd24ce2b6fd518a3cf253ad1ef409388a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:63.698ex; height:5.343ex;" alt="{\displaystyle \pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots }" /> The following table compares the convergence rates of these two series: <table class="wikitable" style="text-align: center; margin: auto;"> <tbody><tr> <th>Infinite series for π</th> <th>After 1st term</th> <th>After 2nd term</th> <th>After 3rd term</th> <th>After 4th term</th> <th>After 5th term</th> <th>Converges to: </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>11</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316235e9f73d29062935aefcb128b9be8060b5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.352ex; height:5.343ex;" alt="{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }" /> </td> <td>4.0000</td> <td>2.6666 ...</td> <td>3.4666 ...</td> <td>2.8952 ...</td> <td>3.3396 ...</td> <td rowspan="2">π = 3.1415 ... </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>4</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>6</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758c0411cbd41eac7aef1ee7c9a46e84d64210b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.691ex; height:5.343ex;" alt="{\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }" /> </td> <td>3.0000</td> <td>3.1666 ...</td> <td>3.1333 ...</td> <td>3.1452 ...</td> <td>3.1396 ... </td></tr></tbody></table> After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include <a href="/wiki/Machin-like_formula" title="Machin-like formula">Machin's series</a> and <a href="/wiki/Chudnovsky_algorithm" title="Chudnovsky algorithm">Chudnovsky's series</a>, the latter producing 14 correct decimal digits per term.<a href="#cite_note-Aconverge-93">[90]</a> <div class="mw-heading mw-heading3"><h3 id="Irrationality_and_transcendence">Irrationality and transcendence</h3></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/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">Proof that π is irrational</a> and <a href="/wiki/Proof_that_%CF%80_is_transcendental" class="mw-redirect" title="Proof that π is transcendental">Proof that π is transcendental</a></div> Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a> in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> that later contributed to the development and study of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>:<a href="#cite_note-Posamentier-98">[95]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf829fbf86ae73080bce0a95c497001b8d15fb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.619ex; height:6.176ex;" alt="{\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }" /> Swiss scientist <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> in 1768 proved that π is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>, meaning it is not equal to the quotient of any two integers.<a href="#cite_note-FOOTNOTEArndtHaenel20065-23">[21]</a> <a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">Lambert's proof</a> exploited a continued-fraction representation of the tangent function.<a href="#cite_note-99">[96]</a> French mathematician <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> proved in 1794 that π2 is also irrational. In 1882, German mathematician <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Ferdinand von Lindemann</a> proved that π is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>,<a href="#cite_note-100">[97]</a> confirming a conjecture made by both <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> and Euler.<a href="#cite_note-FOOTNOTEArndtHaenel2006196-101">[98]</a><a href="#cite_note-102">[99]</a> Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".<a href="#cite_note-103">[100]</a> <div class="mw-heading mw-heading3"><h3 id="Adoption_of_the_symbol_π">Adoption of the symbol π</h3></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:145px;max-width:145px"><div class="thumbimage" style="height:181px;overflow:hidden"><a href="/wiki/File:William_Jones,_the_Mathematician.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/William_Jones%2C_the_Mathematician.jpg/143px-William_Jones%2C_the_Mathematician.jpg" decoding="async" width="143" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/William_Jones%2C_the_Mathematician.jpg/215px-William_Jones%2C_the_Mathematician.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/William_Jones%2C_the_Mathematician.jpg/286px-William_Jones%2C_the_Mathematician.jpg 2x" data-file-width="632" data-file-height="800" /></a></div><div class="thumbcaption">The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician <a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> in 1706</div></div><div class="tsingle" style="width:143px;max-width:143px"><div class="thumbimage" style="height:181px;overflow:hidden"><a href="/wiki/File:Leonhard_Euler.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/250px-Leonhard_Euler.jpg" decoding="async" width="141" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/330px-Leonhard_Euler.jpg 2x" data-file-width="4672" data-file-height="6040" /></a></div><div class="thumbcaption"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> popularized the use of the Greek letter π in works he published in 1736 and 1748.</div></div></div></div></div> The first recorded use of the symbol π in circle geometry is in <a href="/wiki/William_Oughtred" title="William Oughtred">Oughtred's</a> Clavis Mathematicae (1648),<a href="#cite_note-104">[101]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006166-105">[102]</a> where the <a href="/wiki/Greek_letters" class="mw-redirect" title="Greek letters">Greek letters</a> π and δ were combined into the fraction ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>δ</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{\delta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07042fa248fd5ceaffdd918e7291ab50ca86bf79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{\delta }}}" />⁠ for denoting the ratios <a href="/wiki/Semiperimeter" title="Semiperimeter">semiperimeter</a> to <a href="/wiki/Semidiameter" class="mw-redirect" title="Semidiameter">semidiameter</a> and perimeter to diameter, that is, what is presently denoted as π.<a href="#cite_note-firstPi-9">[8]</a><a href="#cite_note-Cajori-2007-106">[103]</a><a href="#cite_note-Smith-1958-107">[104]</a><a href="#cite_note-108">[105]</a> (Before then, mathematicians sometimes used letters such as c or p instead.<a href="#cite_note-FOOTNOTEArndtHaenel2006166-105">[102]</a>) <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Barrow</a> likewise used the same notation,<a href="#cite_note-109">[106]</a> while <a href="/wiki/David_Gregory_(mathematician)" title="David Gregory (mathematician)">Gregory</a> instead used <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\pi }{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62f00c9d88b9d6b7d4401227a9289982b4820a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\textstyle {\frac {\pi }{\rho }}}" /> to represent 6.28... .<a href="#cite_note-110">[107]</a><a href="#cite_note-Smith-1958-107">[104]</a> The earliest known use of the Greek letter π alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician <a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.<a href="#cite_note-jones-4">[3]</a><a href="#cite_note-FOOTNOTEArndtHaenel2006165-111">[108]</a> The Greek letter appears on p. 243 in the phrase "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2558d30e006729499201220e98f83888914bef12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\textstyle {\tfrac {1}{2}}}" /> Periphery (π)", calculated for a circle with radius one. However, Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. <a href="/wiki/John_Machin" title="John Machin">John Machin</a>", leading to speculation that Machin may have employed the Greek letter before Jones.<a href="#cite_note-FOOTNOTEArndtHaenel2006166-105">[102]</a> Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.<a href="#cite_note-Cajori-2007-106">[103]</a><a href="#cite_note-112">[109]</a> <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used π = 6.28..., the ratio of periphery to radius, in this and some later writing.<a href="#cite_note-113">[110]</a><a href="#cite_note-114">[111]</a> Euler first used π = 3.14... in his 1736 work <a href="/wiki/Mechanica" title="Mechanica">Mechanica</a>,<a href="#cite_note-115">[112]</a> and continued in his widely read 1748 work <a href="/wiki/Introductio_in_analysin_infinitorum" title="Introductio in analysin infinitorum">Introductio in analysin infinitorum</a> (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1").<a href="#cite_note-116">[113]</a> Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the <a href="/wiki/Western_world" title="Western world">Western world</a>,<a href="#cite_note-FOOTNOTEArndtHaenel2006166-105">[102]</a> though the definition still varied between 3.14... and 6.28... as late as 1761.<a href="#cite_note-117">[114]</a> <div class="mw-heading mw-heading2"><h2 id="Modern_quest_for_more_digits">Modern quest for more digits</h2></div> <div class="mw-heading mw-heading3"><h3 id="Computer_era_and_iterative_algorithms">Computer era and iterative algorithms</h3></div> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote floatright" style="; font-size: 90%;"> <blockquote class="quotebox-quote left-aligned" style=""> The <a href="/wiki/Gauss%E2%80%93Legendre_algorithm" title="Gauss–Legendre algorithm">Gauss–Legendre iterative algorithm</a>: Initialize <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a_{0}=1,\quad b_{0}={\frac {1}{\sqrt {2}}},\quad t_{0}={\frac {1}{4}},\quad p_{0}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle a_{0}=1,\quad b_{0}={\frac {1}{\sqrt {2}}},\quad t_{0}={\frac {1}{4}},\quad p_{0}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289b9555fe4184d8c18ce84e6a20078fcf53cc07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.574ex; height:4.176ex;" alt="{\displaystyle \textstyle a_{0}=1,\quad b_{0}={\frac {1}{\sqrt {2}}},\quad t_{0}={\frac {1}{4}},\quad p_{0}=1.}" /> Iterate <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a_{n+1}={\frac {a_{n}+b_{n}}{2}},\quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mspace width="1em"></mspace> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle a_{n+1}={\frac {a_{n}+b_{n}}{2}},\quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1078754ad2314049f7046646f743377e6aaf0c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.996ex; height:3.843ex;" alt="{\displaystyle \textstyle a_{n+1}={\frac {a_{n}+b_{n}}{2}},\quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}},}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\quad \quad p_{n+1}=2p_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em"></mspace> <mspace width="1em"></mspace> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\quad \quad p_{n+1}=2p_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab091570d1e7c23425a7c86b021c310ec7f489ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.708ex; height:3.009ex;" alt="{\displaystyle \textstyle t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\quad \quad p_{n+1}=2p_{n}.}" /> Then an estimate for π is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6a29dbd699744dac1aa6c8e615551a94c25b07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.808ex; height:4.843ex;" alt="{\displaystyle \textstyle \pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.}" /> </blockquote> </div> The development of computers in the mid-20th century again revolutionized the hunt for digits of π. Mathematicians <a href="/wiki/John_Wrench" title="John Wrench">John Wrench</a> and Levi Smith reached 1,120 digits in 1949 using a desk calculator.<a href="#cite_note-FOOTNOTEArndtHaenel2006205-118">[115]</a> Using an <a href="/wiki/Inverse_tangent" class="mw-redirect" title="Inverse tangent">inverse tangent</a> (arctan) infinite series, a team led by George Reitwiesner and <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a> computer.<a href="#cite_note-FOOTNOTEArndtHaenel2006197-119">[116]</a><a href="#cite_note-120">[117]</a> The record, always relying on an arctan series, was broken repeatedly (3089 digits in 1955,<a href="#cite_note-121">[118]</a> 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.<a href="#cite_note-FOOTNOTEArndtHaenel2006197-119">[116]</a> Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new <a href="/wiki/Iterative_algorithm" class="mw-redirect" title="Iterative algorithm">iterative algorithms</a> for computing π, which were much faster than the infinite series; and second, the invention of <a href="/wiki/Multiplication_algorithm#Fast_multiplication_algorithms_for_large_inputs" title="Multiplication algorithm">fast multiplication algorithms</a> that could multiply large numbers very rapidly.<a href="#cite_note-FOOTNOTEArndtHaenel200615–17-122">[119]</a> Such algorithms are particularly important in modern π computations because most of the computer's time is devoted to multiplication.<a href="#cite_note-FOOTNOTEArndtHaenel2006131-123">[120]</a> They include the <a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba algorithm</a>, <a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook multiplication</a>, and <a href="/wiki/FFT_multiplication#Fourier_transform_methods" class="mw-redirect" title="FFT multiplication">Fourier transform-based methods</a>.<a href="#cite_note-FOOTNOTEArndtHaenel2006132,_140-124">[121]</a> The iterative algorithms were independently published in 1975–1976 by physicist <a href="/wiki/Eugene_Salamin_(mathematician)" title="Eugene Salamin (mathematician)">Eugene Salamin</a> and scientist <a href="/wiki/Richard_Brent_(scientist)" class="mw-redirect" title="Richard Brent (scientist)">Richard Brent</a>.<a href="#cite_note-FOOTNOTEArndtHaenel200687-125">[122]</a> These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, in what is now termed the <a href="/wiki/AGM_method" class="mw-redirect" title="AGM method">arithmetic–geometric mean method</a> (AGM method) or <a href="/wiki/Gauss%E2%80%93Legendre_algorithm" title="Gauss–Legendre algorithm">Gauss–Legendre algorithm</a>.<a href="#cite_note-FOOTNOTEArndtHaenel200687-125">[122]</a> As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm. The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step. For example, the Brent–Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers <a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">John</a> and <a href="/wiki/Peter_Borwein" title="Peter Borwein">Peter Borwein</a> produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.<a href="#cite_note-126">[123]</a> Iterative methods were used by Japanese mathematician <a href="/wiki/Yasumasa_Kanada" title="Yasumasa Kanada">Yasumasa Kanada</a> to set several records for computing π between 1995 and 2002.<a href="#cite_note-Background-127">[124]</a> This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.<a href="#cite_note-Background-127">[124]</a> <div class="mw-heading mw-heading3"><h3 id="Motives_for_computing_π">Motives for computing π</h3></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Record_pi_approximations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Record_pi_approximations.svg/300px-Record_pi_approximations.svg.png" decoding="async" width="300" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Record_pi_approximations.svg/450px-Record_pi_approximations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Record_pi_approximations.svg/600px-Record_pi_approximations.svg.png 2x" data-file-width="1024" data-file-height="528" /></a><figcaption>As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically. The vertical scale is <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic</a>.</figcaption></figure> For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most <a href="/wiki/Cosmological" class="mw-redirect" title="Cosmological">cosmological</a> calculations, because that is the accuracy necessary to calculate the circumference of the <a href="/wiki/Observable_universe" title="Observable universe">observable universe</a> with a precision of one atom. Accounting for additional digits needed to compensate for computational <a href="/wiki/Round-off_error" title="Round-off error">round-off errors</a>, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute π to thousands and millions of digits.<a href="#cite_note-128">[125]</a> This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.<a href="#cite_note-MSNBC-129">[126]</a><a href="#cite_note-independent.co.uk-130">[127]</a> They also have practical benefits, such as testing <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a>, testing numerical analysis algorithms (including <a href="/wiki/Multiplication_algorithm#Fast_multiplication_algorithms_for_large_inputs" title="Multiplication algorithm">high-precision multiplication algorithms</a>); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.<a href="#cite_note-FOOTNOTEArndtHaenel200618-131">[128]</a> <div class="mw-heading mw-heading3"><h3 id="Rapidly_convergent_series">Rapidly convergent series</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Srinivasa_Ramanujan_-_OPC_-_2_(cleaned).jpg" class="mw-file-description"><img alt="Photo portrait of a man" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Srinivasa_Ramanujan_-_OPC_-_2_%28cleaned%29.jpg/250px-Srinivasa_Ramanujan_-_OPC_-_2_%28cleaned%29.jpg" decoding="async" width="170" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Srinivasa_Ramanujan_-_OPC_-_2_%28cleaned%29.jpg/255px-Srinivasa_Ramanujan_-_OPC_-_2_%28cleaned%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/70/Srinivasa_Ramanujan_-_OPC_-_2_%28cleaned%29.jpg 2x" data-file-width="288" data-file-height="375" /></a><figcaption> <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a>, working in isolation in India, produced many innovative series for computing π.</figcaption></figure> Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.<a href="#cite_note-Background-127">[124]</a> The fast iterative algorithms were anticipated in 1914, when Indian mathematician <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth and rapid convergence.<a href="#cite_note-132">[129]</a> One of his formulae, based on <a href="/wiki/Modular_equation" title="Modular equation">modular equations</a>, is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{k!^{4}\left(396^{4k}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>9801</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mn>1103</mn> <mo>+</mo> <mn>26390</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mn>396</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{k!^{4}\left(396^{4k}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0993bfb34b91c895c4a55899e71b875dc8d4dd3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.518ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{k!^{4}\left(396^{4k}\right)}}.}" /> This series converges much more rapidly than most arctan series, including Machin's formula.<a href="#cite_note-133">[130]</a> <a href="/wiki/Bill_Gosper" title="Bill Gosper">Bill Gosper</a> was the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.<a href="#cite_note-134">[131]</a> Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (<a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Jonathan</a> and <a href="/wiki/Peter_Borwein" title="Peter Borwein">Peter</a>) and the <a href="/wiki/Chudnovsky_brothers" title="Chudnovsky brothers">Chudnovsky brothers</a>.<a href="#cite_note-135">[132]</a> The <a href="/wiki/Chudnovsky_algorithm" title="Chudnovsky algorithm">Chudnovsky formula</a> developed in 1987 is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!\,k!^{3}(-640320)^{3k}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>10005</mn> </msqrt> <mn>4270934400</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mn>13591409</mn> <mo>+</mo> <mn>545140134</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="thinmathspace"></mspace> <mi>k</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mn>640320</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!\,k!^{3}(-640320)^{3k}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ab7f20d9e679a0ab580eb0d1e63bfd7ea0767a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.792ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!\,k!^{3}(-640320)^{3k}}}.}" /> It produces about 14 digits of π per term<a href="#cite_note-136">[133]</a> and has been used for several record-setting π calculations, including the first to surpass 1 billion (109) digits in 1989 by the Chudnovsky brothers, 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo,<a href="#cite_note-NW-137">[134]</a> and 100 trillion digits by <a href="/wiki/Emma_Haruka_Iwao" title="Emma Haruka Iwao">Emma Haruka Iwao</a> in 2022.<a href="#cite_note-138">[135]</a> For similar formulae, see also the <a href="/wiki/Ramanujan%E2%80%93Sato_series" title="Ramanujan–Sato series">Ramanujan–Sato series</a>. In 2006, mathematician <a href="/wiki/Simon_Plouffe" title="Simon Plouffe">Simon Plouffe</a> used the PSLQ <a href="/wiki/Integer_relation_algorithm" title="Integer relation algorithm">integer relation algorithm</a><a href="#cite_note-139">[136]</a> to generate several new formulae for π, conforming to the following template: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948b735a39a1bbf9a39ba0b6bd7dc63bddcd4351" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.161ex; height:6.843ex;" alt="{\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right),}" /> where q is <a href="/wiki/Gelfond%27s_constant" title="Gelfond's constant">eπ</a> (Gelfond's constant), k is an <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd number</a>, and a, b, c are certain rational numbers that Plouffe computed.<a href="#cite_note-140">[137]</a> <div class="mw-heading mw-heading3"><h3 id="Monte_Carlo_methods">Monte Carlo methods</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:217px;max-width:217px"><div class="trow"><div class="tsingle" style="width:133px;max-width:133px"><div class="thumbimage" style="height:78px;overflow:hidden"><a href="/wiki/File:Buffon_needle.svg" class="mw-file-description"><img alt="Needles of length ℓ scattered on stripes with width t" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Buffon_needle.svg/131px-Buffon_needle.svg.png" decoding="async" width="131" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Buffon_needle.svg/197px-Buffon_needle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Buffon_needle.svg/262px-Buffon_needle.svg.png 2x" data-file-width="220" data-file-height="132" /></a></div><div class="thumbcaption"><a href="/wiki/Buffon%27s_needle" class="mw-redirect" title="Buffon's needle">Buffon's needle</a>. Needles a and b are dropped randomly.</div></div><div class="tsingle" style="width:80px;max-width:80px"><div class="thumbimage" style="height:78px;overflow:hidden"><a href="/wiki/File:Pi_30K.gif" class="mw-file-description"><img alt="Thousands of dots randomly covering a square and a circle inscribed in the square." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Pi_30K.gif/120px-Pi_30K.gif" decoding="async" width="78" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Pi_30K.gif/250px-Pi_30K.gif 2x" data-file-width="500" data-file-height="500" /></a></div><div class="thumbcaption">Random dots are placed on a square and a circle inscribed inside.</div></div></div></div></div> <a href="/wiki/Monte_Carlo_methods" class="mw-redirect" title="Monte Carlo methods">Monte Carlo methods</a>, which evaluate the results of multiple random trials, can be used to create approximations of π.<a href="#cite_note-141">[138]</a> <a href="/wiki/Buffon%27s_needle" class="mw-redirect" title="Buffon's needle">Buffon's needle</a> is one such technique: If a needle of length ℓ is dropped n times on a surface on which parallel lines are drawn t units apart, and if x of those times it comes to rest crossing a line (x > 0), then one may approximate π based on the counts:<a href="#cite_note-bn-142">[139]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \approx {\frac {2n\ell }{xt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>ℓ</mi> </mrow> <mrow> <mi>x</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \approx {\frac {2n\ell }{xt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e72a1af2e84df27e3b491d5ffcf140960d0b3af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.44ex; height:5.343ex;" alt="{\displaystyle \pi \approx {\frac {2n\ell }{xt}}.}" /> Another Monte Carlo method for computing π is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal π/4.<a href="#cite_note-143">[140]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Five_random_walks.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/e/e4/Five_random_walks.png/220px-Five_random_walks.png" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e4/Five_random_walks.png/330px-Five_random_walks.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e4/Five_random_walks.png/440px-Five_random_walks.png 2x" data-file-width="732" data-file-height="445" /></a><figcaption>Five random walks with 200 steps. The sample mean of |W200| is μ = 56/5, and so 2(200)μ−2 ≈ 3.19 is within 0.05 of π.</figcaption></figure> Another way to calculate π using probability is to start with a <a href="/wiki/Random_walk" title="Random walk">random walk</a>, generated by a sequence of (fair) coin tosses: independent <a href="/wiki/Random_variable" title="Random variable">random variables</a> Xk such that Xk ∈ {−1,1} with equal probabilities. The associated random walk is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}=\sum _{k=1}^{n}X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}=\sum _{k=1}^{n}X_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff50676493705f91fc45cb5ab160083bbf7aa669" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.266ex; height:6.843ex;" alt="{\displaystyle W_{n}=\sum _{k=1}^{n}X_{k}}" /> so that, for each n, Wn is drawn from a shifted and scaled <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>. As n varies, Wn defines a (discrete) <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>. Then π can be calculated by<a href="#cite_note-144">[141]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{n\to \infty }{\frac {2n}{E[|W_{n}|]^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{n\to \infty }{\frac {2n}{E[|W_{n}|]^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a691be63815c6b7d9fe15070ae98039d9c1d0384" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.402ex; height:6.009ex;" alt="{\displaystyle \pi =\lim _{n\to \infty }{\frac {2n}{E[|W_{n}|]^{2}}}.}" /> This Monte Carlo method is independent of any relation to circles, and is a consequence of the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, discussed <a href="#Gaussian_integrals">below</a>. These Monte Carlo methods for approximating π are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate π when speed or accuracy is desired.<a href="#cite_note-145">[142]</a> <div class="mw-heading mw-heading3"><h3 id="Spigot_algorithms">Spigot algorithms</h3></div> Two algorithms were discovered in 1995 that opened up new avenues of research into π. They are called <a href="/wiki/Spigot_algorithm" title="Spigot algorithm">spigot algorithms</a> because, like water dripping from a <a href="/wiki/Tap_(valve)" title="Tap (valve)">spigot</a>, they produce single digits of π that are not reused after they are calculated.<a href="#cite_note-FOOTNOTEArndtHaenel200677–84-146">[143]</a><a href="#cite_note-Gibbons-147">[144]</a> This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.<a href="#cite_note-FOOTNOTEArndtHaenel200677–84-146">[143]</a> Mathematicians <a href="/wiki/Stan_Wagon" title="Stan Wagon">Stan Wagon</a> and Stanley Rabinowitz produced a simple spigot algorithm in 1995.<a href="#cite_note-Gibbons-147">[144]</a><a href="#cite_note-FOOTNOTEArndtHaenel200677-148">[145]</a><a href="#cite_note-149">[146]</a> Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.<a href="#cite_note-FOOTNOTEArndtHaenel200677-148">[145]</a> Another spigot algorithm, the <a href="/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula" title="Bailey–Borwein–Plouffe formula">BBP</a> <a href="/wiki/Digit_extraction_algorithm" class="mw-redirect" title="Digit extraction algorithm">digit extraction algorithm</a>, was discovered in 1995 by Simon Plouffe:<a href="#cite_note-FOOTNOTEArndtHaenel2006117,_126–128-150">[147]</a><a href="#cite_note-bbpf-151">[148]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</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> <msup> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c092c788f1ff39174a3f78c5a329d5aa83ca2add" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.636ex; height:7.009ex;" alt="{\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}" /> This formula, unlike others before it, can produce any individual <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> digit of π without calculating all the preceding digits.<a href="#cite_note-FOOTNOTEArndtHaenel2006117,_126–128-150">[147]</a> Individual binary digits may be extracted from individual hexadecimal digits, and <a href="/wiki/Octal" title="Octal">octal</a> digits can be extracted from one or two hexadecimal digits. An important application of digit extraction algorithms is to validate new claims of record π computations: After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several randomly selected hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct.<a href="#cite_note-NW-137">[134]</a> Between 1998 and 2000, the <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project <a href="/wiki/PiHex" title="PiHex">PiHex</a> used <a href="/wiki/Bellard%27s_formula" title="Bellard's formula">Bellard's formula</a> (a modification of the BBP algorithm) to compute the quadrillionth (1015th) bit of π, which turned out to be 0.<a href="#cite_note-152">[149]</a> In September 2010, a <a href="/wiki/Yahoo!" class="mw-redirect" title="Yahoo!">Yahoo!</a> employee used the company's <a href="/wiki/Apache_Hadoop" title="Apache Hadoop">Hadoop</a> application on one thousand computers over a 23-day period to compute 256 <a href="/wiki/Bit" title="Bit">bits</a> of π at the two-quadrillionth (2×1015th) bit, which also happens to be zero.<a href="#cite_note-153">[150]</a> In 2022, Plouffe found a base-10 algorithm for calculating digits of π.<a href="#cite_note-154">[151]</a> <div class="mw-heading mw-heading2"><h2 id="Role_and_characterizations_in_mathematics">Role and characterizations in mathematics</h2></div> Because π is closely related to the circle, it is found in <a href="/wiki/List_of_formulae_involving_%CF%80" title="List of formulae involving π">many formulae</a> from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, and number theory, also include π in some of their important formulae. <div class="mw-heading mw-heading3"><h3 id="Geometry_and_trigonometry">Geometry and trigonometry</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_Area.svg" class="mw-file-description"><img alt="A diagram of a circle with a square coving the circle's upper right quadrant." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/220px-Circle_Area.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/330px-Circle_Area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/440px-Circle_Area.svg.png 2x" data-file-width="264" data-file-height="264" /></a><figcaption>The area of the circle equals π times the shaded area. The area of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is π.</figcaption></figure> <a href="#cite_note-155">[152]</a>π appears in formulae for areas and volumes of geometrical shapes based on circles, such as <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, <a href="/wiki/Sphere" title="Sphere">spheres</a>, <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cones</a>, and <a href="/wiki/Torus" title="Torus">tori</a>. Below are some of the more common formulae that involve π.<a href="#cite_note-156">[153]</a> <ul><li>The circumference of a circle with radius r is 2πr.<a href="#cite_note-157">[154]</a></li> <li>The <a href="/wiki/Area_of_a_disk" class="mw-redirect" title="Area of a disk">area of a circle</a> with radius r is πr2.</li> <li>The area of an ellipse with semi-major axis a and semi-minor axis b is πab.<a href="#cite_note-158">[155]</a></li> <li>The volume of a sphere with radius r is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠4/3⁠πr3.</li> <li>The surface area of a sphere with radius r is 4πr2.</li></ul> Some of the formulae above are special cases of the volume of the <a href="/wiki/N-ball" class="mw-redirect" title="N-ball">n-dimensional ball</a> and the surface area of its boundary, the <a href="/wiki/N-sphere" title="N-sphere">(n−1)-dimensional sphere</a>, given <a href="#The_gamma_function_and_Stirling's_approximation">below</a>. Apart from circles, there are other <a href="/wiki/Curve_of_constant_width" title="Curve of constant width">curves of constant width</a>. By <a href="/wiki/Barbier%27s_theorem" title="Barbier's theorem">Barbier's theorem</a>, every curve of constant width has perimeter π times its width. The <a href="/wiki/Reuleaux_triangle" title="Reuleaux triangle">Reuleaux triangle</a> (formed by the intersection of three circles with the sides of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular <a href="/wiki/Smoothness" title="Smoothness">smooth</a> and even <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a> of constant width.<a href="#cite_note-159">[156]</a> <a href="/wiki/Integral" title="Integral">Definite integrals</a> that describe circumference, area, or volume of shapes generated by circles typically have values that involve π. For example, an integral that specifies half the area of a circle of radius one is given by:<a href="#cite_note-160">[157]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1955a15a053adcb91a83af56ea651085c2c18353" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.569ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}" /> In that integral, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875fba356d7f4befcacf6573012100b3ef95cd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.711ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1-x^{2}}}}" /> represents the height over the <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}" />-axis of a <a href="/wiki/Semicircle" title="Semicircle">semicircle</a> (the <a href="/wiki/Square_root" title="Square root">square root</a> is a consequence of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>), and the integral computes the area below the semicircle. The existence of such integrals makes π an <a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">algebraic period</a>.<a href="#cite_note-161">[158]</a> <div class="mw-heading mw-heading3"><h3 id="Units_of_angle">Units of angle</h3></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/Units_of_angle_measure" class="mw-redirect" title="Units of angle measure">Units of angle measure</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Sine_cosine_one_period.svg" class="mw-file-description"><img alt="Diagram showing graphs of functions" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/500px-Sine_cosine_one_period.svg.png" decoding="async" width="340" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/510px-Sine_cosine_one_period.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/680px-Sine_cosine_one_period.svg.png 2x" data-file-width="600" data-file-height="240" /></a><figcaption><a href="/wiki/Sine" class="mw-redirect" title="Sine">Sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> functions repeat with period 2π.</figcaption></figure>The <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> rely on angles, and mathematicians generally use <a href="/wiki/Radian" title="Radian">radians</a> as units of measurement. π plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2π radians. The angle measure of 180° is equal to π radians, and 1° = π/180 radians.<a href="#cite_note-FOOTNOTEAbramson2014[httpsopenstaxorgbooksprecalculuspages5-1-angles_Section_5.1:_Angles]-162">[159]</a> Common trigonometric functions have periods that are multiples of π; for example, sine and cosine have period 2π,<a href="#cite_note-WCS-163">[160]</a> so for any angle θ and any integer k,<a href="#cite_note-WCS-163">[160]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right){\text{ and }}\cos \theta =\cos \left(\theta +2\pi k\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right){\text{ and }}\cos \theta =\cos \left(\theta +2\pi k\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b3c88ab8201f4f0c6f18dd73cb951ebcd07e2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.92ex; height:2.843ex;" alt="{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right){\text{ and }}\cos \theta =\cos \left(\theta +2\pi k\right).}" /> <div class="mw-heading mw-heading3"><h3 id="Eigenvalues">Eigenvalues</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_partials_on_strings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/220px-Harmonic_partials_on_strings.svg.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/330px-Harmonic_partials_on_strings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/440px-Harmonic_partials_on_strings.svg.png 2x" data-file-width="620" data-file-height="590" /></a><figcaption>The <a href="/wiki/Overtone" title="Overtone">overtones</a> of a vibrating string are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the second derivative, and form a <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic progression</a>. The associated eigenvalues form the <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> of integer multiples of π.</figcaption></figure> Many of the appearances of π in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However, π also appears in many natural situations having apparently nothing to do with geometry. In many applications, it plays a distinguished role as an <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a>. For example, an idealized <a href="/wiki/Vibrating_string" class="mw-redirect" title="Vibrating string">vibrating string</a> can be modelled as the graph of a function f on the unit interval [0, 1], with <a href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">fixed ends</a> f(0) = f(1) = 0. The modes of vibration of the string are solutions of the <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)+\lambda f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)+\lambda f(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd94cf1cb3f339145da51a6494cf8ea95c5644c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.471ex; height:3.009ex;" alt="{\displaystyle f''(x)+\lambda f(x)=0}" />, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(t)=-\lambda f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(t)=-\lambda f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bfe6a78ecda33af55f241344a0ce2cc5862794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.786ex; height:3.009ex;" alt="{\displaystyle f''(t)=-\lambda f(x)}" />. Thus λ is an eigenvalue of the second derivative <a href="/wiki/Differential_operator" title="Differential operator">operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto f''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d92f8b043c42cc84705fe283f2e325b8df6b20a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.35ex; height:2.843ex;" alt="{\displaystyle f\mapsto f''}" />, and is constrained by <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a> to take on only certain specific values. It must be positive, since the operator is <a href="/wiki/Negative_definite" class="mw-redirect" title="Negative definite">negative definite</a>, so it is convenient to write λ = ν2, where ν > 0 is called the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a>. Then f(x) = sin(π x) satisfies the boundary conditions and the differential equation with ν = π.<a href="#cite_note-164">[161]</a> The value π is, in fact, the least such value of the wavenumber, and is associated with the <a href="/wiki/Fundamental_mode" class="mw-redirect" title="Fundamental mode">fundamental mode</a> of vibration of the string. One way to show this is by estimating the <a href="/wiki/Energy" title="Energy">energy</a>, which satisfies <a href="/wiki/Wirtinger%27s_inequality_for_functions" title="Wirtinger's inequality for functions">Wirtinger's inequality</a>:<a href="#cite_note-FOOTNOTEDymMcKean197247-165">[162]</a> for a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:[0,1]\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:[0,1]\to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0468f15485d405d64092878cda0fc0cbdab2f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.16ex; height:2.843ex;" alt="{\displaystyle f:[0,1]\to \mathbb {C} }" /> with f(0) = f(1) = 0 and f, f′ both <a href="/wiki/Square_integrable" class="mw-redirect" title="Square integrable">square integrable</a>, we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{2}\int _{0}^{1}|f(x)|^{2}\,dx\leq \int _{0}^{1}|f'(x)|^{2}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{2}\int _{0}^{1}|f(x)|^{2}\,dx\leq \int _{0}^{1}|f'(x)|^{2}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9021233d9bdca65d39dc801347f5901f04e55204" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.455ex; height:6.176ex;" alt="{\displaystyle \pi ^{2}\int _{0}^{1}|f(x)|^{2}\,dx\leq \int _{0}^{1}|f'(x)|^{2}\,dx,}" /> with equality precisely when f is a multiple of sin(π x). Here π appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the <a href="/wiki/Variational_theorem" class="mw-redirect" title="Variational theorem">variational characterization</a> of the eigenvalue. As a consequence, π is the smallest <a href="/wiki/Singular_value" title="Singular value">singular value</a> of the derivative operator on the space of functions on [0, 1] vanishing at both endpoints (the <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}^{1}[0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7113d194de39d54621e9da47782ad5263b9f1790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.81ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}[0,1]}" />). <div class="mw-heading mw-heading3"><h3 id="Inequalities">Inequalities</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Sir_William_Thompson_illustration_of_Carthage.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Sir_William_Thompson_illustration_of_Carthage.png/220px-Sir_William_Thompson_illustration_of_Carthage.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d8/Sir_William_Thompson_illustration_of_Carthage.png 1.5x" data-file-width="286" data-file-height="287" /></a><figcaption>The <a href="/wiki/Ancient_Carthage" title="Ancient Carthage">ancient city of Carthage</a> was the solution to an isoperimetric problem, according to a legend recounted by <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>:<a href="#cite_note-166">[163]</a> those lands bordering the sea that <a href="/wiki/Dido" title="Dido">Queen Dido</a> could enclose on all other sides within a single given oxhide, cut into strips.</figcaption></figure> The number π serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned <a href="#Definition">above</a>, it can be characterized via its role as the best constant in the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a>: the area A enclosed by a plane <a href="/wiki/Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a> of perimeter P satisfies the inequality <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi A\leq P^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mi>A</mi> <mo>≤</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi A\leq P^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da151eb3d06c6538dfa721d85c36e2b052a5114" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.859ex; height:3.009ex;" alt="{\displaystyle 4\pi A\leq P^{2},}" /> and equality is clearly achieved for the circle, since in that case A = πr2 and P = 2πr.<a href="#cite_note-167">[164]</a> Ultimately, as a consequence of the isoperimetric inequality, π appears in the optimal constant for the critical <a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a> in n dimensions, which thus characterizes the role of π in many physical phenomena as well, for example those of classical <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a>.<a href="#cite_note-168">[165]</a><a href="#cite_note-169">[166]</a><a href="#cite_note-170">[167]</a> In two dimensions, the critical Sobolev inequality is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \|f\|_{2}\leq \|\nabla f\|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \|f\|_{2}\leq \|\nabla f\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80eaaf07fddda0c919000c07fda359208e10cf9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.844ex; height:2.843ex;" alt="{\displaystyle 2\pi \|f\|_{2}\leq \|\nabla f\|_{1}}" /> for f a smooth function with compact support in R2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b4d6de89b52c5a5e6e1583cb63eaee263e307b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \nabla f}" /> is the <a href="/wiki/Gradient" title="Gradient">gradient</a> of f, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0c64fd6ef8f74eac7ace4d9b3db3f4d148385d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.658ex; height:2.843ex;" alt="{\displaystyle \|f\|_{2}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\nabla f\|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\nabla f\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de8dc3d0605c979ebae27c6791e3df7b1c19887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.594ex; height:2.843ex;" alt="{\displaystyle \|\nabla f\|_{1}}" /> refer respectively to the <a href="/wiki/Lp_space" title="Lp space">L2 and L1-norm</a>. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants. Wirtinger's inequality also generalizes to higher-dimensional <a href="/wiki/Poincar%C3%A9_inequality" title="Poincaré inequality">Poincaré inequalities</a> that provide best constants for the <a href="/wiki/Dirichlet_energy" title="Dirichlet energy">Dirichlet energy</a> of an n-dimensional membrane. Specifically, π is the greatest constant such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \leq {\frac {\left(\int _{G}|\nabla u|^{2}\right)^{1/2}}{\left(\int _{G}|u|^{2}\right)^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \leq {\frac {\left(\int _{G}|\nabla u|^{2}\right)^{1/2}}{\left(\int _{G}|u|^{2}\right)^{1/2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b72bae66777c4041fd32d69be2a43a740bd885" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:19.362ex; height:11.176ex;" alt="{\displaystyle \pi \leq {\frac {\left(\int _{G}|\nabla u|^{2}\right)^{1/2}}{\left(\int _{G}|u|^{2}\right)^{1/2}}}}" /> for all <a href="/wiki/Convex_set" title="Convex set">convex</a> subsets G of Rn of diameter 1, and square-integrable functions u on G of mean zero.<a href="#cite_note-171">[168]</a> Just as Wirtinger's inequality is the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">variational</a> form of the <a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a> problem in one dimension, the Poincaré inequality is the variational form of the <a href="/wiki/Neumann_problem" class="mw-redirect" title="Neumann problem">Neumann</a> eigenvalue problem, in any dimension. <div class="mw-heading mw-heading3"><h3 id="Fourier_transform_and_Heisenberg_uncertainty_principle">Fourier transform and Heisenberg uncertainty principle</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Animation_of_Heisenberg_geodesic.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Animation_of_Heisenberg_geodesic.gif/220px-Animation_of_Heisenberg_geodesic.gif" decoding="async" width="220" height="302" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/00/Animation_of_Heisenberg_geodesic.gif 1.5x" data-file-width="315" data-file-height="432" /></a><figcaption>An animation of a <a href="/wiki/Heisenberg_group#As_a_sub-Riemannian_manifold" title="Heisenberg group">geodesic in the Heisenberg group</a></figcaption></figure> The constant π also appears as a critical spectral parameter in the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>. This is the <a href="/wiki/Integral_transform" title="Integral transform">integral transform</a>, that takes a complex-valued integrable function f on the real line to the function defined as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a78cc05a0945443f61906a7c66822e38839e84f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.062ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx.}" /> Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve π somewhere. The above is the most canonical definition, however, giving the unique unitary operator on L2 that is also an algebra homomorphism of L1 to L∞.<a href="#cite_note-172">[169]</a> The <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a> also contains the number π. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq \left({\frac {1}{4\pi }}\int _{-\infty }^{\infty }|f(x)|^{2}\,dx\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mo>≥</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq \left({\frac {1}{4\pi }}\int _{-\infty }^{\infty }|f(x)|^{2}\,dx\right)^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ed97af513c67ad7eb0f44a51c76696255f16ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.056ex; height:6.509ex;" alt="{\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq \left({\frac {1}{4\pi }}\int _{-\infty }^{\infty }|f(x)|^{2}\,dx\right)^{2}.}" /> The physical consequence, about the uncertainty in simultaneous position and momentum observations of a <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> system, is <a href="#Describing_physical_phenomena">discussed below</a>. The appearance of π in the formulae of Fourier analysis is ultimately a consequence of the <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>, asserting the uniqueness of the <a href="/wiki/Schr%C3%B6dinger_representation" class="mw-redirect" title="Schrödinger representation">Schrödinger representation</a> of the <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a>.<a href="#cite_note-howe-173">[170]</a> <div class="mw-heading mw-heading3"><h3 id="Gaussian_integrals">Gaussian integrals</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:E%5E(-x%5E2).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/220px-E%5E%28-x%5E2%29.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/330px-E%5E%28-x%5E2%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/E%5E%28-x%5E2%29.svg/440px-E%5E%28-x%5E2%29.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>A graph of the <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a> ƒ(x) = e−x2. The coloured region between the function and the x-axis has area √π.</figcaption></figure> The fields of <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a> frequently use the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.<a href="#cite_note-174">[171]</a> The <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a>, which is the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of the normal distribution with <a href="/wiki/Mean" title="Mean">mean</a> μ and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> σ, naturally contains π:<a href="#cite_note-GaussProb-175">[172]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824d182749fb4097ca7bf06c7435c066756892db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.757ex; height:6.176ex;" alt="{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}" /> The factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef676eb866fd962ce8fe724b168c57499b2d8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.969ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}" /> makes the area under the graph of f equal to one, as is required for a probability distribution. This follows from a <a href="/wiki/Integration_by_substitution" title="Integration by substitution">change of variables</a> in the <a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a>:<a href="#cite_note-GaussProb-175">[172]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\,du={\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\,du={\sqrt {\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3890ce940b1efd925d23c908144db3c4049dc8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.499ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\,du={\sqrt {\pi }}}" /> which says that the area under the basic <a href="/wiki/Bell_curve" class="mw-redirect" title="Bell curve">bell curve</a> in the figure is equal to the square root of π. The <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> explains the central role of normal distributions, and thus of π, in probability and statistics. This theorem is ultimately connected with the <a href="#Fourier_transform_and_Heisenberg_uncertainty_principle">spectral characterization</a> of π as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.<a href="#cite_note-FOOTNOTEDymMcKean1972Section_2.7-176">[173]</a> Equivalently, π is the unique constant making the Gaussian normal distribution e−πx2 equal to its own Fourier transform.<a href="#cite_note-177">[174]</a> Indeed, according to <a href="#CITEREFHowe1980">Howe (1980)</a>, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.<a href="#cite_note-howe-173">[170]</a> <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Order-7_triangular_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/220px-Order-7_triangular_tiling.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/330px-Order-7_triangular_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/440px-Order-7_triangular_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a><figcaption><a href="/wiki/Uniformization_theorem" title="Uniformization theorem">Uniformization</a> of the <a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a>, a surface of <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> three and Euler characteristic −4, as a quotient of the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">hyperbolic plane</a> by the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> <a href="/wiki/PSL(2,7)" title="PSL(2,7)">PSL(2,7)</a> of the <a href="/wiki/Fano_plane" title="Fano plane">Fano plane</a>. The hyperbolic area of a fundamental domain is 8π, by Gauss–Bonnet.</figcaption></figure> The constant π appears in the <a href="/wiki/Gauss%E2%80%93Bonnet_formula" class="mw-redirect" title="Gauss–Bonnet formula">Gauss–Bonnet formula</a> which relates the <a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">differential geometry of surfaces</a> to their <a href="/wiki/Topology" title="Topology">topology</a>. Specifically, if a <a href="/wiki/Compact_space" title="Compact space">compact</a> surface Σ has <a href="/wiki/Gauss_curvature" class="mw-redirect" title="Gauss curvature">Gauss curvature</a> K, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Σ</mi> </mrow> </msub> <mi>K</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>χ</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c846ab9590f095656ecf41398c7d91b9f3a00e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.046ex; height:5.676ex;" alt="{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma )}" /> where χ(Σ) is the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a>, which is an integer.<a href="#cite_note-178">[175]</a> An example is the surface area of a sphere S of curvature 1 (so that its <a href="/wiki/Radius_of_curvature" title="Radius of curvature">radius of curvature</a>, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology groups</a> and is found to be equal to two. Thus we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(S)=\int _{S}1\,dA=2\pi \cdot 2=4\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(S)=\int _{S}1\,dA=2\pi \cdot 2=4\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f541185019255fc11e6ef909d681e99b9490cf88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.658ex; height:5.676ex;" alt="{\displaystyle A(S)=\int _{S}1\,dA=2\pi \cdot 2=4\pi }" /> reproducing the formula for the surface area of a sphere of radius 1. The constant appears in many other integral formulae in topology, in particular, those involving <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic classes</a> via the <a href="/wiki/Chern%E2%80%93Weil_homomorphism" title="Chern–Weil homomorphism">Chern–Weil homomorphism</a>.<a href="#cite_note-179">[176]</a> <div class="mw-heading mw-heading3"><h3 id="Cauchy's_integral_formula">Cauchy's integral formula</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Factorial05.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Factorial05.jpg/220px-Factorial05.jpg" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Factorial05.jpg/330px-Factorial05.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Factorial05.jpg/440px-Factorial05.jpg 2x" data-file-width="760" data-file-height="598" /></a><figcaption>Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles. Here illustrated is the complex logarithm of the Gamma function.</figcaption></figure> One of the key tools in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> is <a href="/wiki/Contour_integration" title="Contour integration">contour integration</a> of a function over a positively oriented (<a href="/wiki/Rectifiable_curve" class="mw-redirect" title="Rectifiable curve">rectifiable</a>) <a href="/wiki/Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a> γ. A form of <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> states that if a point z0 is interior to γ, then<a href="#cite_note-180">[177]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\gamma }{\frac {dz}{z-z_{0}}}=2\pi i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\gamma }{\frac {dz}{z-z_{0}}}=2\pi i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17569b627f1050cac4952724e6e56209e44dc425" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.747ex; height:6.176ex;" alt="{\displaystyle \oint _{\gamma }{\frac {dz}{z-z_{0}}}=2\pi i.}" /> Although the curve γ is not a circle, and hence does not have any obvious connection to the constant π, a standard proof of this result uses <a href="/wiki/Morera%27s_theorem" title="Morera's theorem">Morera's theorem</a>, which implies that the integral is invariant under <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curve γ does not contain z0, then the above integral is 2πi times the <a href="/wiki/Winding_number" title="Winding number">winding number</a> of the curve. The general form of Cauchy's integral formula establishes the relationship between the values of a <a href="/wiki/Complex_analytic_function" class="mw-redirect" title="Complex analytic function">complex analytic function</a> f(z) on the Jordan curve γ and the value of f(z) at any interior point z0 of γ:<a href="#cite_note-181">[178]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\gamma }{f(z) \over z-z_{0}}\,dz=2\pi if(z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\gamma }{f(z) \over z-z_{0}}\,dz=2\pi if(z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f239d18251cef9208918cba90587fe2d7f618ae2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.014ex; height:6.509ex;" alt="{\displaystyle \oint _{\gamma }{f(z) \over z-z_{0}}\,dz=2\pi if(z_{0})}" /> provided f(z) is analytic in the region enclosed by γ and extends continuously to γ. Cauchy's integral formula is a special case of the <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>, that if g(z) is a <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic function</a> the region enclosed by γ and is continuous in a neighbourhood of γ, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\gamma }g(z)\,dz=2\pi i\sum \operatorname {Res} (g,a_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>∑</mo> <mi>Res</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\gamma }g(z)\,dz=2\pi i\sum \operatorname {Res} (g,a_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fffa5445c67252fba1e631674f6202925ebe3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.971ex; height:6.009ex;" alt="{\displaystyle \oint _{\gamma }g(z)\,dz=2\pi i\sum \operatorname {Res} (g,a_{k})}" /> where the sum is of the <a href="/wiki/Residue_(mathematics)" class="mw-redirect" title="Residue (mathematics)">residues</a> at the <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of g(z). <div class="mw-heading mw-heading3"><h3 id="Vector_calculus_and_physics">Vector calculus and physics</h3></div> The constant π is ubiquitous in <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> and <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a>, for example in <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a>,<a href="#cite_note-182">[179]</a> <a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss's law</a>, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, and even the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>.<a href="#cite_note-183">[180]</a><a href="#cite_note-184">[181]</a> Perhaps the simplest example of this is the two-dimensional <a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a>, representing the potential of a point source at the origin, whose associated field has unit outward <a href="/wiki/Flux" title="Flux">flux</a> through any smooth and oriented closed surface enclosing the source: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\mathbf {x} )={\frac {1}{2\pi }}\log |\mathbf {x} |.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {x} )={\frac {1}{2\pi }}\log |\mathbf {x} |.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173b53b5c1644e0cd4d59433349d3faf3ac98d72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.425ex; height:5.176ex;" alt="{\displaystyle \Phi (\mathbf {x} )={\frac {1}{2\pi }}\log |\mathbf {x} |.}" /> The factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d777af7ccaf5b0339d298cedf9757fe87586da1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle 1/2\pi }" /> is necessary to ensure that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" /> is the <a href="/wiki/Fundamental_solution" title="Fundamental solution">fundamental solution</a> of the <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" />:<a href="#cite_note-Elliptic_PDE2-185">[182]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \Phi =\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Phi =\delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7b32c7fab3ef26726f4d1040a0e553d1e99eb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.761ex; height:2.343ex;" alt="{\displaystyle \Delta \Phi =\delta }" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" /> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. In higher dimensions, factors of π are present because of a normalization by the n-dimensional volume of the unit <a href="/wiki/N_sphere" class="mw-redirect" title="N sphere">n sphere</a>. For example, in three dimensions, the Newtonian potential is:<a href="#cite_note-Elliptic_PDE2-185">[182]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\mathbf {x} )=-{\frac {1}{4\pi |\mathbf {x} |}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {x} )=-{\frac {1}{4\pi |\mathbf {x} |}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d621580a0535c73a9a18911166a0c95010d47c3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.487ex; height:6.009ex;" alt="{\displaystyle \Phi (\mathbf {x} )=-{\frac {1}{4\pi |\mathbf {x} |}},}" /> which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator. <div class="mw-heading mw-heading3"><h3 id="Total_curvature">Total curvature</h3></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Total_curvature" title="Total curvature">Total curvature</a>.[<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Total_curvature&action=edit">edit</a>]</div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Winding_Number_Around_Point.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Winding_Number_Around_Point.svg/300px-Winding_Number_Around_Point.svg.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Winding_Number_Around_Point.svg/450px-Winding_Number_Around_Point.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Winding_Number_Around_Point.svg/600px-Winding_Number_Around_Point.svg.png 2x" data-file-width="380" data-file-height="280" /></a><figcaption>This curve has total curvature 6π, and index/turning number 3, though it only has <a href="/wiki/Winding_number" title="Winding number">winding number</a> 2 about p.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> study of the <a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">differential geometry of curves</a>, the <a href="/wiki/Total_curvature" title="Total curvature">total curvature</a> of an <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersed</a> <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> is the <a href="/wiki/Integral" title="Integral">integral</a> of <a href="/wiki/Curvature" title="Curvature">curvature</a> along a curve taken with respect to <a href="/wiki/Arc_length" title="Arc length">arc length</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>k</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4185b690e5ee1729126fe5ad581607e83a908ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.897ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.}" /></dd></dl> The total curvature of a closed curve is always an integer multiple of 2π, where N is called the <a href="/wiki/Index_of_the_curve" class="mw-redirect" title="Index of the curve">index of the curve</a> or <a href="/wiki/Turning_number" class="mw-redirect" title="Turning number">turning number</a> – it is the <a href="/wiki/Winding_number" title="Winding number">winding number</a> of the unit <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> about the origin, or equivalently the degree of the map to the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the <a href="/wiki/Gauss_map" title="Gauss map">Gauss map</a> for surfaces.</div></div> <div class="mw-heading mw-heading3"><h3 id="The_gamma_function_and_Stirling's_approximation">The gamma function and Stirling's approximation</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_plot_points_marked.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Gamma_plot_points_marked.svg/220px-Gamma_plot_points_marked.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Gamma_plot_points_marked.svg/330px-Gamma_plot_points_marked.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Gamma_plot_points_marked.svg/440px-Gamma_plot_points_marked.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Plot of the gamma function on the real axis</figcaption></figure> The <a href="/wiki/Factorial" title="Factorial">factorial</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}" /> is the product of all of the positive integers through n. The <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> extends the concept of <a href="/wiki/Factorial" title="Factorial">factorial</a> (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (n)=(n-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (n)=(n-1)!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3f7eebd96f717c5f1fd154b3905af7fbcabf24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.609ex; height:2.843ex;" alt="{\displaystyle \Gamma (n)=(n-1)!}" />. When the gamma function is evaluated at half-integers, the result contains π. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75337a521eb1fcd407b39db28c38ca6f559ab3b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.116ex; height:3.009ex;" alt="{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5c2d3a05e28b445968213c15aecbee42ce3221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.817ex; height:4.176ex;" alt="{\textstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}" />.<a href="#cite_note-186">[183]</a> The gamma function is defined by its <a href="/wiki/Weierstrass_product" class="mw-redirect" title="Weierstrass product">Weierstrass product</a> development:<a href="#cite_note-187">[184]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }{\frac {e^{z/n}}{1+z/n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> <mi>z</mi> </mrow> </msup> <mi>z</mi> </mfrac> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }{\frac {e^{z/n}}{1+z/n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39107a18b7680b95e8d2af238137b27fe1221168" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.886ex; height:6.843ex;" alt="{\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }{\frac {e^{z/n}}{1+z/n}}}" /> where γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. Evaluated at z = 1/2 and squared, the equation Γ(1/2)2 = π reduces to the Wallis product formula. The gamma function is also connected to the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and identities for the <a href="/wiki/Functional_determinant" title="Functional determinant">functional determinant</a>, in which the constant π <a href="#Number_theory_and_Riemann_zeta_function">plays an important role</a>. The gamma function is used to calculate the volume Vn(r) of the <a href="/wiki/N-ball" class="mw-redirect" title="N-ball">n-dimensional ball</a> of radius r in Euclidean n-dimensional space, and the surface area Sn−1(r) of its boundary, the <a href="/wiki/N-sphere" title="N-sphere">(n−1)-dimensional sphere</a>:<a href="#cite_note-188">[185]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}(r)={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}(r)={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62d4adfe7ec97afd2e3fca97bc77971419b3d30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.075ex; height:7.009ex;" alt="{\displaystyle V_{n}(r)={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n-1}(r)={\frac {n\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n-1}(r)={\frac {n\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cb69ab6975502f9d348cc6ca9a128c8c96a47e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.346ex; height:7.009ex;" alt="{\displaystyle S_{n-1}(r)={\frac {n\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n-1}.}" /> Further, it follows from the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi r={\frac {S_{n+1}(r)}{V_{n}(r)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r={\frac {S_{n+1}(r)}{V_{n}(r)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807a85f4bb84937de20521b0db1fc83295a48532" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.726ex; height:6.509ex;" alt="{\displaystyle 2\pi r={\frac {S_{n+1}(r)}{V_{n}(r)}}.}" /> The gamma function can be used to create a simple approximation to the factorial function n! for large n: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 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/4f32274cb48ebb4eb4b2379884758a61f17c05db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.136ex; height:3.343ex;" alt="{\textstyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}" /> which is known as <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a>.<a href="#cite_note-189">[186]</a> Equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\lim _{n\to \infty }{\frac {e^{2n}n!^{2}}{2n^{2n+1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mi>n</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\lim _{n\to \infty }{\frac {e^{2n}n!^{2}}{2n^{2n+1}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bedea644b29f119cf4d2175576c12d61010d01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.271ex; height:6.009ex;" alt="{\displaystyle \pi =\lim _{n\to \infty }{\frac {e^{2n}n!^{2}}{2n^{2n+1}}}.}" /> As a geometrical application of Stirling's approximation, let Δn denote the <a href="/wiki/Simplex" title="Simplex">standard simplex</a> in n-dimensional Euclidean space, and (n + 1)Δn denote the simplex having all of its sides scaled up by a factor of n + 1. Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Vol} ((n+1)\Delta _{n})={\frac {(n+1)^{n}}{n!}}\sim {\frac {e^{n+1}}{\sqrt {2\pi n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Vol} ((n+1)\Delta _{n})={\frac {(n+1)^{n}}{n!}}\sim {\frac {e^{n+1}}{\sqrt {2\pi n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae31c667990be5b0307c3ab5233509497d33bac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.489ex; height:6.676ex;" alt="{\displaystyle \operatorname {Vol} ((n+1)\Delta _{n})={\frac {(n+1)^{n}}{n!}}\sim {\frac {e^{n+1}}{\sqrt {2\pi n}}}.}" /> <a href="/wiki/Ehrhart%27s_volume_conjecture" title="Ehrhart's volume conjecture">Ehrhart's volume conjecture</a> is that this is the (optimal) upper bound on the volume of a <a href="/wiki/Convex_body" title="Convex body">convex body</a> containing only one <a href="/wiki/Lattice_point" class="mw-redirect" title="Lattice point">lattice point</a>.<a href="#cite_note-190">[187]</a> <div class="mw-heading mw-heading3"><h3 id="Number_theory_and_Riemann_zeta_function">Number theory and Riemann zeta function</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pr%C3%BCfer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pr%C3%BCfer.png/220px-Pr%C3%BCfer.png" decoding="async" width="220" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pr%C3%BCfer.png/330px-Pr%C3%BCfer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pr%C3%BCfer.png/440px-Pr%C3%BCfer.png 2x" data-file-width="1255" data-file-height="901" /></a><figcaption>Each prime has an associated <a href="/wiki/Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>, which are arithmetic localizations of the circle. The <a href="/wiki/L-function" title="L-function">L-functions</a> of analytic number theory are also localized in each prime p.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ModularGroup-FundamentalDomain.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/250px-ModularGroup-FundamentalDomain.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/330px-ModularGroup-FundamentalDomain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/ModularGroup-FundamentalDomain.svg/440px-ModularGroup-FundamentalDomain.svg.png 2x" data-file-width="401" data-file-height="181" /></a><figcaption>Solution of the Basel problem using the <a href="/wiki/Weil_conjecture_on_Tamagawa_numbers" class="mw-redirect" title="Weil conjecture on Tamagawa numbers">Weil conjecture</a>: the value of ζ(2) is the <a href="/wiki/Poincar%C3%A9_half-plane_model" title="Poincaré half-plane model">hyperbolic</a> area of a fundamental domain of the <a href="/wiki/Modular_group" title="Modular group">modular group</a>, times π/2.</figcaption></figure> The <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> ζ(s) is used in many areas of mathematics. When evaluated at s = 2 it can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581714c279e80acf87edfb221004f575d4770e97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.568ex; height:5.676ex;" alt="{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }" /> Finding a <a href="/wiki/Closed-form_expression" title="Closed-form expression">simple solution</a> for this infinite series was a famous problem in mathematics called the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a>. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> solved it in 1735 when he showed it was equal to π2/6.<a href="#cite_note-Posamentier-98">[95]</a> Euler's result leads to the <a href="/wiki/Number_theory" title="Number theory">number theory</a> result that the probability of two random numbers being <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a> (that is, having no shared factors) is equal to 6/π2.<a href="#cite_note-191">[188]</a><a href="#cite_note-192">[189]</a> This probability is based on the observation that the probability that any number is <a href="/wiki/Divisible" class="mw-redirect" title="Divisible">divisible</a> by a prime p is 1/p (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is 1/p2, and the probability that at least one of them is not is 1 − 1/p2. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:<a href="#cite_note-193">[190]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)&=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}\\[4pt]&={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}\\[4pt]&={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.\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="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>61</mn> <mi mathvariant="normal">%</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)&=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}\\[4pt]&={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}\\[4pt]&={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ad11b6609d91487577949c7a42872afdc33a36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:36.922ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}\prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)&=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}\\[4pt]&={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}\\[4pt]&={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.\end{aligned}}}" /> This probability can be used in conjunction with a <a href="/wiki/Random_number_generator" class="mw-redirect" title="Random number generator">random number generator</a> to approximate π using a Monte Carlo approach.<a href="#cite_note-194">[191]</a> The solution to the Basel problem implies that the geometrically derived quantity π is connected in a deep way to the distribution of prime numbers. This is a special case of <a href="/wiki/Weil%27s_conjecture_on_Tamagawa_numbers" title="Weil's conjecture on Tamagawa numbers">Weil's conjecture on Tamagawa numbers</a>, which asserts the equality of similar such infinite products of arithmetic quantities, localized at each prime p, and a geometrical quantity: the reciprocal of the volume of a certain <a href="/wiki/Locally_symmetric_space" class="mw-redirect" title="Locally symmetric space">locally symmetric space</a>. In the case of the Basel problem, it is the <a href="/wiki/Hyperbolic_3-manifold" title="Hyperbolic 3-manifold">hyperbolic 3-manifold</a> <a href="/wiki/SL2(R)" title="SL2(R)">SL2(R)</a>/<a href="/wiki/Modular_group" title="Modular group">SL2(Z)</a>.<a href="#cite_note-195">[192]</a> The zeta function also satisfies Riemann's functional equation, which involves π as well as the gamma function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>s</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b4eab44e38b188e9cdde9f8d8344ca79a980ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.716ex; height:4.843ex;" alt="{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).}" /> Furthermore, the derivative of the zeta function satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(-\zeta '(0))={\sqrt {2\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(-\zeta '(0))={\sqrt {2\pi }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2907a3d1a8927b4fd2a276f0463569cc1f77b6f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.12ex; height:3.176ex;" alt="{\displaystyle \exp(-\zeta '(0))={\sqrt {2\pi }}.}" /> A consequence is that π can be obtained from the <a href="/wiki/Functional_determinant" title="Functional determinant">functional determinant</a> of the <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a>. This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.<a href="#cite_note-196">[193]</a> The calculation can be recast in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, specifically the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">variational approach</a> to the <a href="/wiki/Bohr_model" title="Bohr model">spectrum of the hydrogen atom</a>.<a href="#cite_note-197">[194]</a> <div class="mw-heading mw-heading3"><h3 id="Fourier_series">Fourier series</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:2-adic_integers_with_dual_colorings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/2-adic_integers_with_dual_colorings.svg/220px-2-adic_integers_with_dual_colorings.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/2-adic_integers_with_dual_colorings.svg/330px-2-adic_integers_with_dual_colorings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/2-adic_integers_with_dual_colorings.svg/440px-2-adic_integers_with_dual_colorings.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>π appears in characters of <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">p-adic numbers</a> (shown), which are elements of a <a href="/wiki/Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>. <a href="/wiki/Tate%27s_thesis" title="Tate's thesis">Tate's thesis</a> makes heavy use of this machinery.<a href="#cite_note-198">[195]</a></figcaption></figure> The constant π also appears naturally in <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> of <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a>. Periodic functions are functions on the group T =R/Z of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function f on T can be written as an infinite linear superposition of <a href="/wiki/Unitary_character" class="mw-redirect" title="Unitary character">unitary characters</a> of T. That is, continuous <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphisms</a> from T to the <a href="/wiki/Circle_group" title="Circle group">circle group</a> U(1) of unit modulus complex numbers. It is a theorem that every character of T is one of the complex exponentials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{n}(x)=e^{2\pi inx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{n}(x)=e^{2\pi inx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142bcbcd2610f5640cdfd50166a105624164d007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.113ex; height:3.176ex;" alt="{\displaystyle e_{n}(x)=e^{2\pi inx}}" />. There is a unique character on T, up to complex conjugation, that is a group isomorphism. Using the <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a> on the circle group, the constant π is half the magnitude of the <a href="/wiki/Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2π.<a href="#cite_note-Nicolas_Bourbaki-22">[20]</a> As a result, the constant π is the unique number such that the group T, equipped with its Haar measure, is <a href="/wiki/Pontrjagin_dual" class="mw-redirect" title="Pontrjagin dual">Pontrjagin dual</a> to the <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> of integral multiples of 2π.<a href="#cite_note-FOOTNOTEDymMcKean1972Chapter_4-199">[196]</a> This is a version of the one-dimensional <a href="/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a>. <div class="mw-heading mw-heading3"><h3 id="Modular_forms_and_theta_functions">Modular forms and theta functions</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lattice_with_tau.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/c/ca/Lattice_with_tau.svg/220px-Lattice_with_tau.svg.png" decoding="async" width="220" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/c/ca/Lattice_with_tau.svg/330px-Lattice_with_tau.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/c/ca/Lattice_with_tau.svg/440px-Lattice_with_tau.svg.png 2x" data-file-width="158" data-file-height="134" /></a><figcaption>Theta functions transform under the <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> of periods of an elliptic curve.</figcaption></figure> The constant π is connected in a deep way with the theory of <a href="/wiki/Modular_form" title="Modular form">modular forms</a> and <a href="/wiki/Theta_function" title="Theta function">theta functions</a>. For example, the <a href="/wiki/Chudnovsky_algorithm" title="Chudnovsky algorithm">Chudnovsky algorithm</a> involves in an essential way the <a href="/wiki/J-invariant" title="J-invariant">j-invariant</a> of an <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curve</a>. <a href="/wiki/Modular_form" title="Modular form">Modular forms</a> are <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> in the <a href="/wiki/Upper_half_plane" class="mw-redirect" title="Upper half plane">upper half plane</a> characterized by their transformation properties under the <a href="/wiki/Modular_group" title="Modular group">modular group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2d95ba975aa85bd6c46f0a696872d3aa385869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.159ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}" /> (or its various subgroups), a lattice in the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb95adfe7c34f38de771bb5fa44e17ec26deeee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.287ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}" />. An example is the <a href="/wiki/Jacobi_theta_function" class="mw-redirect" title="Jacobi theta function">Jacobi theta function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty }e^{2\pi inz\ +\ \pi in^{2}\tau }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>z</mi> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mi>π</mi> <mi>i</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty }e^{2\pi inz\ +\ \pi in^{2}\tau }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f50d5773dc1a50d50774df58ca32b47d507f26" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.915ex; height:6.843ex;" alt="{\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty }e^{2\pi inz\ +\ \pi in^{2}\tau }}" /> which is a kind of modular form called a <a href="/wiki/Jacobi_form" title="Jacobi form">Jacobi form</a>.<a href="#cite_note-Mumford_1983_1–117-200">[197]</a> This is sometimes written in terms of the <a href="/wiki/Nome_(mathematics)" title="Nome (mathematics)">nome</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{\pi i\tau }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{\pi i\tau }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04b5fc2797318028727bd06733bb52300c99e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.843ex; height:3.009ex;" alt="{\displaystyle q=e^{\pi i\tau }}" />. The constant π is the unique constant making the Jacobi theta function an <a href="/wiki/Automorphic_form" title="Automorphic form">automorphic form</a>, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\theta (z,\tau ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>τ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mi>τ</mi> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\theta (z,\tau ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df3eda20c1a72c699a40cf8b4e0ef49801f601" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.568ex; height:3.176ex;" alt="{\displaystyle \theta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\theta (z,\tau ),}" /> which implies that θ transforms as a representation under the discrete <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a>. General modular forms and other <a href="/wiki/Theta_function" title="Theta function">theta functions</a> also involve π, once again because of the <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>.<a href="#cite_note-Mumford_1983_1–117-200">[197]</a> <div class="mw-heading mw-heading3"><h3 id="Cauchy_distribution_and_potential_theory">Cauchy distribution and potential theory</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Witch_of_Agnesi,_construction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Witch_of_Agnesi%2C_construction.svg/220px-Witch_of_Agnesi%2C_construction.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Witch_of_Agnesi%2C_construction.svg/330px-Witch_of_Agnesi%2C_construction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Witch_of_Agnesi%2C_construction.svg/440px-Witch_of_Agnesi%2C_construction.svg.png 2x" data-file-width="398" data-file-height="223" /></a><figcaption>The <a href="/wiki/Witch_of_Agnesi" title="Witch of Agnesi">Witch of Agnesi</a>, named for <a href="/wiki/Maria_Gaetana_Agnesi" title="Maria Gaetana Agnesi">Maria Agnesi</a> (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:2d_random_walk_ag_adatom_ag111.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/2d_random_walk_ag_adatom_ag111.gif/220px-2d_random_walk_ag_adatom_ag111.gif" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/2d_random_walk_ag_adatom_ag111.gif/330px-2d_random_walk_ag_adatom_ag111.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/2d_random_walk_ag_adatom_ag111.gif/440px-2d_random_walk_ag_adatom_ag111.gif 2x" data-file-width="606" data-file-height="442" /></a><figcaption>The Cauchy distribution governs the passage of <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian particles</a> through a membrane.</figcaption></figure> The <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {1}{\pi }}\cdot {\frac {1}{x^{2}+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {1}{\pi }}\cdot {\frac {1}{x^{2}+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fbbd1733585d1552298a9352029f90887c23a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.424ex; height:5.676ex;" alt="{\displaystyle g(x)={\frac {1}{\pi }}\cdot {\frac {1}{x^{2}+1}}}" /> is a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>. The total probability is equal to one, owing to the integral: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\,dx=\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\,dx=\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f17dafb3b274d126111151564a8666bb4ac93a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.066ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\,dx=\pi .}" /> The <a href="/wiki/Shannon_entropy" class="mw-redirect" title="Shannon entropy">Shannon entropy</a> of the Cauchy distribution is equal to ln(4π), which also involves π. The Cauchy distribution plays an important role in <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a> because it is the simplest <a href="/wiki/Furstenberg_boundary" title="Furstenberg boundary">Furstenberg measure</a>, the classical <a href="/wiki/Poisson_kernel" title="Poisson kernel">Poisson kernel</a> associated with a <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> in a half-plane.<a href="#cite_note-201">[198]</a> <a href="/wiki/Conjugate_harmonic_function" class="mw-redirect" title="Conjugate harmonic function">Conjugate harmonic functions</a> and so also the <a href="/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a> are associated with the asymptotics of the Poisson kernel. The Hilbert transform H is the integral transform given by the <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> of the <a href="/wiki/Singular_integral" title="Singular integral">singular integral</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)\,dx}{x-t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)\,dx}{x-t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd815ba69c72a4cb8e65180491f6db68eda5a12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.312ex; height:6.343ex;" alt="{\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)\,dx}{x-t}}.}" /> The constant π is the unique (positive) normalizing factor such that H defines a <a href="/wiki/Linear_complex_structure" title="Linear complex structure">linear complex structure</a> on the Hilbert space of square-integrable real-valued functions on the real line.<a href="#cite_note-202">[199]</a> The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space L2(R): up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.<a href="#cite_note-203">[200]</a> The constant π is the unique normalizing factor that makes this transformation unitary. <div class="mw-heading mw-heading3"><h3 id="In_the_Mandelbrot_set">In the Mandelbrot set</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg" class="mw-file-description"><img alt="An complex black shape on a blue background." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mandel_zoom_00_mandelbrot_set.jpg/220px-Mandel_zoom_00_mandelbrot_set.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mandel_zoom_00_mandelbrot_set.jpg/330px-Mandel_zoom_00_mandelbrot_set.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mandel_zoom_00_mandelbrot_set.jpg/440px-Mandel_zoom_00_mandelbrot_set.jpg 2x" data-file-width="2560" data-file-height="1920" /></a><figcaption>The <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> can be used to approximate π.</figcaption></figure> An occurrence of π in the <a href="/wiki/Fractal" title="Fractal">fractal</a> called the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> was discovered by David Boll in 1991.<a href="#cite_note-KA-204">[201]</a> He examined the behaviour of the Mandelbrot set near the "neck" at (−0.75, 0). When the number of iterations until divergence for the point (−0.75, ε) is multiplied by ε, the result approaches π as ε approaches zero. The point (0.25 + ε, 0) at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of ε tends to π.<a href="#cite_note-KA-204">[201]</a><a href="#cite_note-205">[202]</a> <div class="mw-heading mw-heading3"><h3 id="Projective_geometry">Projective geometry</h3></div> Let V be the set of all twice differentiable real functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" /> that satisfy the <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)+f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)+f(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d948ffddc3fe33d35ea728ff1b825ad52c52f910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.116ex; height:3.009ex;" alt="{\displaystyle f''(x)+f(x)=0}" />. Then V is a two-dimensional real <a href="/wiki/Vector_space" title="Vector space">vector space</a>, with two parameters corresponding to a pair of <a href="/wiki/Initial_conditions" class="mw-redirect" title="Initial conditions">initial conditions</a> for the differential equation. For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }" />, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{t}:V\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{t}:V\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0893b2e58dd291ddcccec348b735b1b82c665ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.926ex; height:2.509ex;" alt="{\displaystyle e_{t}:V\to \mathbb {R} }" /> be the evaluation functional, which associates to each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd9aeefd10ac4d735121b2f9f0a195a4894dba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.906ex; height:2.509ex;" alt="{\displaystyle f\in V}" /> the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{t}(f)=f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{t}(f)=f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01ae70e278abe3102c5727dd656049fb46067a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.023ex; height:2.843ex;" alt="{\displaystyle e_{t}(f)=f(t)}" /> of the function f at the real point t. Then, for each t, the <a href="/wiki/Kernel_of_a_linear_transformation" class="mw-redirect" title="Kernel of a linear transformation">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0bd4f3c8809bace7f01662b5e10a1cc4aa2d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle e_{t}}" /> is a one-dimensional linear subspace of V. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto \ker e_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <mi>ker</mi> <mo>⁡</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto \ker e_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e918446fd8ebdc1056dd32fedb00cdfffbba127b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.922ex; height:2.509ex;" alt="{\displaystyle t\mapsto \ker e_{t}}" /> defines a function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to \mathbb {P} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {P} (V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce02dea63b500b84bd1f6c7bf4cf975079f345d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {P} (V)}" /> from the real line to the <a href="/wiki/Real_projective_line" title="Real projective line">real projective line</a>. This function is periodic, and the quantity π can be characterized as the period of this map.<a href="#cite_note-206">[203]</a> This is notable in that the constant π, rather than 2π, appears naturally in this context. <div class="mw-heading mw-heading2"><h2 id="Outside_mathematics">Outside mathematics</h2></div> <div class="mw-heading mw-heading3"><h3 id="Describing_physical_phenomena">Describing physical phenomena</h3></div> Although not a <a href="/wiki/Physical_constant" title="Physical constant">physical constant</a>, π appears routinely in equations describing fundamental principles of the universe, often because of π's relationship to the circle and to <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate systems</a>. A simple formula from the field of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> gives the approximate period T of a simple <a href="/wiki/Pendulum" title="Pendulum">pendulum</a> of length L, swinging with a small amplitude (g is the <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">earth's gravitational acceleration</a>):<a href="#cite_note-207">[204]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≈</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>L</mi> <mi>g</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd035c36f4530cfdbe4d650dad9a9fb96600c271" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.619ex; height:7.509ex;" alt="{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}" /> One of the key formulae of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> is <a href="/wiki/Heisenberg%27s_uncertainty_principle" class="mw-redirect" title="Heisenberg's uncertainty principle">Heisenberg's uncertainty principle</a>, which shows that the uncertainty in the measurement of a particle's position (Δx) and <a href="/wiki/Momentum" title="Momentum">momentum</a> (Δp) cannot both be arbitrarily small at the same time (where h is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>):<a href="#cite_note-208">[205]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2897ef997b77cd45b82bf7f217d4a614fefdbfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.834ex; height:5.343ex;" alt="{\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}" /> The fact that π is approximately equal to 3 plays a role in the relatively long lifetime of <a href="/wiki/Orthopositronium" class="mw-redirect" title="Orthopositronium">orthopositronium</a>. The inverse lifetime to lowest order in the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> α is<a href="#cite_note-209">[206]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m_{\text{e}}\alpha ^{6},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> </mrow> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m_{\text{e}}\alpha ^{6},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d51fbd59e6edefd2ddcb3f4843642da06d32de5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.718ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m_{\text{e}}\alpha ^{6},}" /> where me is the mass of the electron. π is present in some structural engineering formulae, such as the <a href="/wiki/Buckling" title="Buckling">buckling</a> formula derived by Euler, which gives the maximum axial load F that a long, slender column of length L, <a href="/wiki/Modulus_of_elasticity" class="mw-redirect" title="Modulus of elasticity">modulus of elasticity</a> E, and <a href="/wiki/Area_moment_of_inertia" class="mw-redirect" title="Area moment of inertia">area moment of inertia</a> I can carry without buckling:<a href="#cite_note-210">[207]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>E</mi> <mi>I</mi> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ec2b531f168b03efdfcdfd27061df942e2a4c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.658ex; height:5.843ex;" alt="{\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}" /> The field of <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> contains π in <a href="/wiki/Stokes%27_law" title="Stokes' law">Stokes' law</a>, which approximates the <a href="/wiki/Drag_force" class="mw-redirect" title="Drag force">frictional force</a> F exerted on small, <a href="/wiki/Sphere" title="Sphere">spherical</a> objects of radius R, moving with velocity v in a <a href="/wiki/Fluid" title="Fluid">fluid</a> with <a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a> η:<a href="#cite_note-211">[208]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=6\pi \eta Rv.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> <mi>η</mi> <mi>R</mi> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=6\pi \eta Rv.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e327b1b8e506d7dd945915675ce1ad287e53cdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.042ex; height:2.676ex;" alt="{\displaystyle F=6\pi \eta Rv.}" /> In electromagnetics, the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a> constant μ0 appears in <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, which describe the properties of <a href="/wiki/Electric_field" title="Electric field">electric</a> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic</a> fields and <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>. Before 20 May 2019, it was defined as exactly <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}\approx 1.2566370614\ldots \times 10^{-6}{\text{ N/A}}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> H/m</mtext> </mrow> <mo>≈</mo> <mn>1.2566370614</mn> <mo>…</mo> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> N/A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}\approx 1.2566370614\ldots \times 10^{-6}{\text{ N/A}}^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8fa061c51aebf2f41aee0bbc787b6f599fd4019" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.04ex; height:3.343ex;" alt="{\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}\approx 1.2566370614\ldots \times 10^{-6}{\text{ N/A}}^{2}.}" /> <div class="mw-heading mw-heading3"><h3 id="Memorizing_digits">Memorizing digits</h3></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/Piphilology" title="Piphilology">Piphilology</a></div> <a href="/wiki/Piphilology" title="Piphilology">Piphilology</a> is the practice of memorizing large numbers of digits of π,<a href="#cite_note-A445-212">[209]</a> and world-records are kept by the <a href="/wiki/Guinness_World_Records" title="Guinness World Records">Guinness World Records</a>. The record for memorizing digits of π, certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015.<a href="#cite_note-213">[210]</a> In 2006, <a href="/wiki/Akira_Haraguchi" title="Akira Haraguchi">Akira Haraguchi</a>, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.<a href="#cite_note-japantimes-214">[211]</a> One common technique is to memorize a story or poem in which the word lengths represent the digits of π: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. Such memorization aids are called <a href="/wiki/Mnemonic" title="Mnemonic">mnemonics</a>. An early example of a mnemonic for pi, originally devised by English scientist <a href="/wiki/James_Hopwood_Jeans" class="mw-redirect" title="James Hopwood Jeans">James Jeans</a>, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."<a href="#cite_note-A445-212">[209]</a> When a poem is used, it is sometimes referred to as a piem.<a href="#cite_note-215">[212]</a> Poems for memorizing π have been composed in several languages in addition to English.<a href="#cite_note-A445-212">[209]</a> Record-setting π memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the <a href="/wiki/Method_of_loci" title="Method of loci">method of loci</a>.<a href="#cite_note-216">[213]</a> A few authors have used the digits of π to establish a new form of <a href="/wiki/Constrained_writing" title="Constrained writing">constrained writing</a>, where the word lengths are required to represent the digits of π. The <a href="/wiki/Cadaeic_Cadenza" class="mw-redirect" title="Cadaeic Cadenza">Cadaeic Cadenza</a> contains the first 3835 digits of π in this manner,<a href="#cite_note-217">[214]</a> and the full-length book Not a Wake contains 10,000 words, each representing one digit of π.<a href="#cite_note-KeithNAW-218">[215]</a> <div class="mw-heading mw-heading3"><h3 id="In_popular_culture">In popular culture</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pi_pie2.jpg" class="mw-file-description"><img alt="Pi Pie at Delft University" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pi_pie2.jpg/220px-Pi_pie2.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pi_pie2.jpg/330px-Pi_pie2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pi_pie2.jpg/440px-Pi_pie2.jpg 2x" data-file-width="577" data-file-height="576" /></a><figcaption>A pi pie. Many <a href="/wiki/Pies" class="mw-redirect" title="Pies">pies</a> are circular, and "pie" and π are <a href="/wiki/Homophones" class="mw-redirect" title="Homophones">homophones</a>, making pie a frequent subject of pi <a href="/wiki/Pun" title="Pun">puns</a>.</figcaption></figure> Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, π has been represented in popular culture more than other mathematical constructs.<a href="#cite_note-219">[216]</a> In the <a href="/wiki/Palais_de_la_D%C3%A9couverte" title="Palais de la Découverte">Palais de la Découverte</a> (a science museum in Paris) there is a circular room known as the pi room. On its wall are inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematician <a href="/wiki/William_Shanks" title="William Shanks">William Shanks</a>, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.<a href="#cite_note-220">[217]</a> In <a href="/wiki/Carl_Sagan" title="Carl Sagan">Carl Sagan</a>'s 1985 novel <a href="/wiki/Contact_(novel)" title="Contact (novel)">Contact</a> it is suggested that the creator of the universe buried a message deep within the digits of π. This part of the story was omitted from the <a href="/wiki/Contact_(1997_American_film)" title="Contact (1997 American film)">film</a> adaptation of the novel.<a href="#cite_note-221">[218]</a><a href="#cite_note-222">[219]</a> The digits of π have also been incorporated into the lyrics of the song "Pi" from the 2005 album <a href="/wiki/Aerial_(album)" title="Aerial (album)">Aerial</a> by <a href="/wiki/Kate_Bush" title="Kate Bush">Kate Bush</a>.<a href="#cite_note-223">[220]</a> In the 1967 <a href="/wiki/Star_Trek:_The_Original_Series" title="Star Trek: The Original Series">Star Trek</a> episode "<a href="/wiki/Wolf_in_the_Fold" title="Wolf in the Fold">Wolf in the Fold</a>", an out-of-control computer is contained by being instructed to "Compute to the last digit the value of π".<a href="#cite_note-life-of-pi-53">[50]</a> In the United States, <a href="/wiki/Pi_Day" title="Pi Day">Pi Day</a> falls on 14 March (written 3/14 in the US style), and is popular among students.<a href="#cite_note-life-of-pi-53">[50]</a> π and its digital representation are often used by self-described "math <a href="/wiki/Geek" title="Geek">geeks</a>" for <a href="/wiki/Inside_joke" class="mw-redirect" title="Inside joke">inside jokes</a> among mathematically and technologically minded groups. A <a href="/wiki/Cheering#Chants_in_North_American_sports" title="Cheering">college cheer</a> variously attributed to the <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a> or the <a href="/wiki/Rensselaer_Polytechnic_Institute" title="Rensselaer Polytechnic Institute">Rensselaer Polytechnic Institute</a> includes "3.14159".<a href="#cite_note-224">[221]</a><a href="#cite_note-225">[222]</a> Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi.<a href="#cite_note-226">[223]</a><a href="#cite_note-227">[224]</a> In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi Approximation Day", as 22/7 = 3.142857.<a href="#cite_note-228">[225]</a> Some have proposed replacing π by <a href="/wiki/Tau_(mathematical_constant)" class="mw-redirect" title="Tau (mathematical constant)">τ = 2π</a>,<a href="#cite_note-229">[226]</a> arguing that τ, as the number of radians in one <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a> or the ratio of a circle's circumference to its radius, is more natural than π and simplifies many formulae.<a href="#cite_note-230">[227]</a><a href="#cite_note-231">[228]</a> This use of τ has not made its way into mainstream mathematics,<a href="#cite_note-232">[229]</a> but since 2010 this has led to people celebrating Two Pi Day or Tau Day on June 28.<a href="#cite_note-233">[230]</a> In 1897, an amateur mathematician attempted to persuade the <a href="/wiki/Indiana_General_Assembly" title="Indiana General Assembly">Indiana legislature</a> to pass the <a href="/wiki/Indiana_Pi_Bill" class="mw-redirect" title="Indiana Pi Bill">Indiana Pi Bill</a>, which described a method to <a href="/wiki/Squaring_the_circle" title="Squaring the circle">square the circle</a> and contained text that implied various incorrect values for π, including 3.2. The bill is notorious as an attempt to establish a value of mathematical constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, and thus it did not become a law.<a href="#cite_note-234">[231]</a> In contemporary <a href="/wiki/Internet_culture" title="Internet culture">internet culture</a>, individuals and organizations frequently pay homage to the number π. For instance, 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/TeX" title="TeX">TeX</a> approach π. The versions are 3, 3.1, 3.14, and so forth.<a href="#cite_note-235">[232]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a></li> <li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">Chronology of computation of π</a></li> <li><a href="/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <div class="mw-heading mw-heading3"><h3 id="Explanatory_notes">Explanatory notes</h3></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> In particular, π is conjectured to be a <a href="/wiki/Normal_number" title="Normal number">normal number</a>, which implies a specific kind of statistical randomness on its digits in all bases. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> The specific integral that Weierstrass used was<a href="#cite_note-14">[13]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55cafb02752f6c0a8f9014c298acc06af20abb96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.134ex; height:6.009ex;" alt="{\displaystyle \pi =\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}.}" /> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> The polynomial shown is the first few terms of the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> function. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEAndrewsAskeyRoy199959-2"><a href="#cite_ref-FOOTNOTEAndrewsAskeyRoy199959_2-0">^</a> <a href="#CITEREFAndrewsAskeyRoy1999">Andrews, Askey & Roy 1999</a>, p. 59. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</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="CITEREFGupta1992" class="citation journal cs1">Gupta, R. C. (1992). "On the remainder term in the Madhava–Leibniz's series". Ganita Bharati. 14 (1–4): 68–71.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ganita+Bharati&rft.atitle=On+the+remainder+term+in+the+Madhava%E2%80%93Leibniz%27s+series&rft.volume=14&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E68-%3C%2Fspan%3E71&rft.date=1992&rft.aulast=Gupta&rft.aufirst=R.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-jones-4">^ <a href="#cite_ref-jones_4-0">a</a> <a href="#cite_ref-jones_4-1">b</a> <a href="#cite_ref-jones_4-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJones1706" class="citation book cs1"><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">Jones, William</a> (1706). <a rel="nofollow" class="external text" href="https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/">Synopsis Palmariorum Matheseos</a>. London: J. Wale. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/">243</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/">263</a>. p. 263: <q>There are various other ways of finding the Lengths, or Areas of particular Curve Lines or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to Circumference as 1 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>239</mn> </mfrac> </mstyle> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <msup> <mn>239</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <msup> <mn>239</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">&</mi> <mi>c</mi> <mo>.</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1223e3144f68554af81860169320f418ec2a106d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:45.778ex; height:4.843ex;" alt="{\displaystyle {\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=}" /> 3.14159, &c. = π. This Series (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. <a href="/wiki/John_Machin" title="John Machin">John Machin</a>; and by means thereof, <a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Van Ceulen</a>'s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Synopsis+Palmariorum+Matheseos&rft.place=London&rft.pages=243%2C+263&rft.pub=J.+Wale&rft.date=1706&rft.aulast=Jones&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FSynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics%2Fpage%2Fn283%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1929" class="citation book cs1">Smith, David Eugene (1929). <a rel="nofollow" class="external text" href="https://archive.org/details/sourcebookinmath1929smit/page/346/">"William Jones: The First Use of π for the Circle Ratio"</a>. A Source Book in Mathematics. McGraw–Hill. pp. 346–347.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=William+Jones%3A+The+First+Use+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E+for+the+Circle+Ratio&rft.btitle=A+Source+Book+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E346-%3C%2Fspan%3E347&rft.pub=McGraw%E2%80%93Hill&rft.date=1929&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsourcebookinmath1929smit%2Fpage%2F346%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.pi2e.ch/">"πe trillion digits of π"</a>. pi2e.ch. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161206063441/http://www.pi2e.ch/">Archived</a> from the original on 6 December 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=pi2e.ch&rft.atitle=%CF%80%3Csup%3Ee%3C%2Fsup%3E+trillion+digits+of+%CF%80&rft_id=http%3A%2F%2Fwww.pi2e.ch%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" 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="CITEREFHaruka_Iwao2019" class="citation web cs1"><a href="/wiki/Emma_Haruka_Iwao" title="Emma Haruka Iwao">Haruka Iwao, Emma</a> (14 March 2019). <a rel="nofollow" class="external text" href="https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud">"Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud"</a>. <a href="/wiki/Google_Cloud_Platform" title="Google Cloud Platform">Google Cloud Platform</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191019023120/https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud">Archived</a> from the original on 19 October 2019. Retrieved 12 April 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Cloud+Platform&rft.atitle=Pi+in+the+sky%3A+Calculating+a+record-breaking+31.4+trillion+digits+of+Archimedes%27+constant+on+Google+Cloud&rft.date=2019-03-14&rft.aulast=Haruka+Iwao&rft.aufirst=Emma&rft_id=https%3A%2F%2Fcloud.google.com%2Fblog%2Fproducts%2Fcompute%2Fcalculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel200617-7"><a href="#cite_ref-FOOTNOTEArndtHaenel200617_7-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 17. </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="CITEREFBaileyPlouffeBorweinBorwein1997" class="citation journal cs1">Bailey, David H.; Plouffe, Simon M.; Borwein, Peter B.; Borwein, Jonathan M. 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Hirzel. p. 195. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160914204826/https://archive.org/details/dieelementederm02baltgoog">Archived</a> from the original on 14 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Elemente+der+Mathematik&rft.pages=195&rft.pub=Hirzel&rft.date=1870&rft.aulast=Baltzer&rft.aufirst=Richard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdieelementederm02baltgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLandau1934" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1934). Einführung in die Differentialrechnung und Integralrechnung (in German). 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Gale Group. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7876-3933-4" title="Special:BookSources/978-0-7876-3933-4"><bdi>978-0-7876-3933-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191213112426/https://archive.org/details/scienceitstimesu0000unse">Archived</a> from the original on 13 December 2019. Retrieved 19 December 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Its+Times%3A+Understanding+the+Social+Significance+of+Scientific+Discovery&rft.pub=Gale+Group&rft.date=2001&rft.isbn=978-0-7876-3933-4&rft.aulast=Schlager&rft.aufirst=Neil&rft.au=Lauer%2C+Josh&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fscienceitstimesu0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988">, p. 185. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMurtyRath2014" class="citation book cs1">Murty, M. Ram; Rath, Purusottam (2014). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-1-4939-0832-5">Transcendental Numbers</a>. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4939-0832-5">10.1007/978-1-4939-0832-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-0831-8" title="Special:BookSources/978-1-4939-0831-8"><bdi>978-1-4939-0831-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transcendental+Numbers&rft.pub=Springer&rft.date=2014&rft_id=info%3Adoi%2F10.1007%2F978-1-4939-0832-5&rft.isbn=978-1-4939-0831-8&rft.aulast=Murty&rft.aufirst=M.+Ram&rft.au=Rath%2C+Purusottam&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-1-4939-0832-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWaldschmidt2021" class="citation web cs1">Waldschmidt, Michel (2021). <a rel="nofollow" class="external text" href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf">"Schanuel's Conjecture: algebraic independence of transcendental numbers"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Schanuel%27s+Conjecture%3A+algebraic+independence+of+transcendental+numbers&rft.date=2021&rft.aulast=Waldschmidt&rft.aufirst=Michel&rft_id=https%3A%2F%2Fwebusers.imj-prg.fr%2F~michel.waldschmidt%2Farticles%2Fpdf%2FSchanuelEn.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Lindemann-WeierstrassTheorem.html">"Lindemann-Weierstrass Theorem"</a>. mathworld.wolfram.com. Retrieved 26 October 2024.</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=Lindemann-Weierstrass+Theorem&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLindemann-WeierstrassTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Eymard_1999_78-37">^ <a href="#cite_ref-Eymard_1999_78_37-0">a</a> <a href="#cite_ref-Eymard_1999_78_37-1">b</a> <a href="#CITEREFEymardLafon2004">Eymard & Lafon 2004</a>, p. 78 </li> <li id="cite_note-FOOTNOTEArndtHaenel200633-38"><a href="#cite_ref-FOOTNOTEArndtHaenel200633_38-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 33. </li> <li id="cite_note-mollin-39">^ <a href="#cite_ref-mollin_39-0">a</a> <a href="#cite_ref-mollin_39-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMollin1999" class="citation journal cs1">Mollin, R. A. (1999). "Continued fraction gems". Nieuw Archief voor Wiskunde. 17 (3): 383–405. <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=1743850">1743850</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nieuw+Archief+voor+Wiskunde&rft.atitle=Continued+fraction+gems&rft.volume=17&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E383-%3C%2Fspan%3E405&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1743850%23id-name%3DMR&rft.aulast=Mollin&rft.aufirst=R.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLange1999" class="citation journal cs1">Lange, L.J. (May 1999). "An Elegant Continued Fraction for π". <a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 106 (5): 456–458. <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%2F2589152">10.2307/2589152</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/2589152">2589152</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+Mathematical+Monthly&rft.atitle=An+Elegant+Continued+Fraction+for+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.volume=106&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E456-%3C%2Fspan%3E458&rft.date=1999-05&rft_id=info%3Adoi%2F10.2307%2F2589152&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589152%23id-name%3DJSTOR&rft.aulast=Lange&rft.aufirst=L.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006240-41"><a href="#cite_ref-FOOTNOTEArndtHaenel2006240_41-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 240. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006242-42"><a href="#cite_ref-FOOTNOTEArndtHaenel2006242_42-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 242. </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKennedy1978" class="citation journal cs1">Kennedy, E.S. (1978). "Abu-r-Raihan al-Biruni, 973–1048". Journal for the History of Astronomy. 9: 65. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978JHA.....9...65K">1978JHA.....9...65K</a>. <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%2F002182867800900106">10.1177/002182867800900106</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:126383231">126383231</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+for+the+History+of+Astronomy&rft.atitle=Abu-r-Raihan+al-Biruni%2C+973%E2%80%931048&rft.volume=9&rft.pages=65&rft.date=1978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126383231%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1177%2F002182867800900106&rft_id=info%3Abibcode%2F1978JHA.....9...65K&rft.aulast=Kennedy&rft.aufirst=E.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a> used a three-sexagesimal-digit approximation, and <a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> expanded this to nine digits; see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAaboe1964" class="citation book cs1"><a href="/wiki/Asger_Aaboe" title="Asger Aaboe">Aaboe, Asger</a> (1964). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA125">Episodes from the Early History of Mathematics</a>. New Mathematical Library. Vol. 13. New York: Random House. p. 125. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-613-0" title="Special:BookSources/978-0-88385-613-0"><bdi>978-0-88385-613-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161129205051/https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA125">Archived</a> from the original on 29 November 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Episodes+from+the+Early+History+of+Mathematics&rft.place=New+York&rft.series=New+Mathematical+Library&rft.pages=125&rft.pub=Random+House&rft.date=1964&rft.isbn=978-0-88385-613-0&rft.aulast=Aaboe&rft.aufirst=Asger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5wGzF0wPFYgC%26pg%3DPA125&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEAbramson2014[httpsopenstaxorgbooksprecalculuspages8-5-polar-form-of-complex-numbers_Section_8.5:_Polar_form_of_complex_numbers]-44"><a href="#cite_ref-FOOTNOTEAbramson2014[httpsopenstaxorgbooksprecalculuspages8-5-polar-form-of-complex-numbers_Section_8.5:_Polar_form_of_complex_numbers]_44-0">^</a> <a href="#CITEREFAbramson2014">Abramson 2014</a>, <a rel="nofollow" class="external text" href="https://openstax.org/books/precalculus/pages/8-5-polar-form-of-complex-numbers">Section 8.5: Polar form of complex numbers</a>. </li> <li id="cite_note-EF-45">^ <a href="#cite_ref-EF_45-0">a</a> <a href="#cite_ref-EF_45-1">b</a> <a href="#CITEREFBronshteĭnSemendiaev1971">Bronshteĭn & Semendiaev 1971</a>, p. 592 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaor2009" class="citation book cs1">Maor, Eli (2009). E: The Story of a Number. Princeton University Press. p. 160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14134-3" title="Special:BookSources/978-0-691-14134-3"><bdi>978-0-691-14134-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=E%3A+The+Story+of+a+Number&rft.pages=160&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-14134-3&rft.aulast=Maor&rft.aufirst=Eli&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEAndrewsAskeyRoy199914-47"><a href="#cite_ref-FOOTNOTEAndrewsAskeyRoy199914_47-0">^</a> <a href="#CITEREFAndrewsAskeyRoy1999">Andrews, Askey & Roy 1999</a>, p. 14. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006167-48">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006167_48-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006167_48-1">b</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 167. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerz-Fischler2000" class="citation book cs1">Herz-Fischler, Roger (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=066T3YLuhA0C&pg=67">The Shape of the Great Pyramid</a>. Wilfrid Laurier University Press. pp. 67–77, 165–166. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88920-324-2" title="Special:BookSources/978-0-88920-324-2"><bdi>978-0-88920-324-2</bdi></a>. Retrieved 5 June 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shape+of+the+Great+Pyramid&rft.pages=%3Cspan+class%3D%22nowrap%22%3E67-%3C%2Fspan%3E77%2C+%3Cspan+class%3D%22nowrap%22%3E165-%3C%2Fspan%3E166&rft.pub=Wilfrid+Laurier+University+Press&rft.date=2000&rft.isbn=978-0-88920-324-2&rft.aulast=Herz-Fischler&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D066T3YLuhA0C%26pg%3D67&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPlofker2009" class="citation book cs1">Plofker, Kim (2009). <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)">Mathematics in India</a>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DHvThPNp9yMC&pg=PA27">27</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691120676" title="Special:BookSources/978-0691120676"><bdi>978-0691120676</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+India&rft.pages=27&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0691120676&rft.aulast=Plofker&rft.aufirst=Kim&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006170-51"><a href="#cite_ref-FOOTNOTEArndtHaenel2006170_51-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 170. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006175,_205-52"><a href="#cite_ref-FOOTNOTEArndtHaenel2006175,_205_52-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 175, 205. </li> <li id="cite_note-life-of-pi-53">^ <a href="#cite_ref-life-of-pi_53-0">a</a> <a href="#cite_ref-life-of-pi_53-1">b</a> <a href="#cite_ref-life-of-pi_53-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBorwein2014" class="citation book cs1"><a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan M.</a> (2014). "The life of π: from Archimedes to ENIAC and beyond". In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren. Heidelberg: Springer. pp. 531–561. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-36736-6_24">10.1007/978-3-642-36736-6_24</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-36735-9" title="Special:BookSources/978-3-642-36735-9"><bdi>978-3-642-36735-9</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=3203895">3203895</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+life+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E%3A+from+Archimedes+to+ENIAC+and+beyond&rft.btitle=From+Alexandria%2C+through+Baghdad%3A+Surveys+and+studies+in+the+ancient+Greek+and+medieval+Islamic+mathematical+sciences+in+honor+of+J.+L.+Berggren&rft.place=Heidelberg&rft.pages=%3Cspan+class%3D%22nowrap%22%3E531-%3C%2Fspan%3E561&rft.pub=Springer&rft.date=2014&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3203895%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-36736-6_24&rft.isbn=978-3-642-36735-9&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006171-54"><a href="#cite_ref-FOOTNOTEArndtHaenel2006171_54-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 171. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006176-55"><a href="#cite_ref-FOOTNOTEArndtHaenel2006176_55-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 176. </li> <li id="cite_note-FOOTNOTEBoyerMerzbach1991168-56"><a href="#cite_ref-FOOTNOTEBoyerMerzbach1991168_56-0">^</a> <a href="#CITEREFBoyerMerzbach1991">Boyer & Merzbach 1991</a>, p. 168. </li> <li id="cite_note-ArPI-57"><a href="#cite_ref-ArPI_57-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 15–16, 175, 184–186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006176–177-58"><a href="#cite_ref-FOOTNOTEArndtHaenel2006176–177_58-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 176–177. </li> <li id="cite_note-autogenerated202-59">^ <a href="#cite_ref-autogenerated202_59-0">a</a> <a href="#cite_ref-autogenerated202_59-1">b</a> <a href="#CITEREFBoyerMerzbach1991">Boyer & Merzbach 1991</a>, p. 202 </li> <li id="cite_note-FOOTNOTEArndtHaenel2006177-60"><a href="#cite_ref-FOOTNOTEArndtHaenel2006177_60-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 177. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006178-61"><a href="#cite_ref-FOOTNOTEArndtHaenel2006178_61-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 178. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006179-62"><a href="#cite_ref-FOOTNOTEArndtHaenel2006179_62-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 179. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006180-63">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006180_63-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006180_63-1">b</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 180. </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAzarian2010" class="citation journal cs1">Azarian, Mohammad K. (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.35834%2Fmjms%2F1312233136">"al-Risāla al-muhītīyya: A Summary"</a>. Missouri Journal of Mathematical Sciences. 22 (2): 64–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.35834%2Fmjms%2F1312233136">10.35834/mjms/1312233136</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Missouri+Journal+of+Mathematical+Sciences&rft.atitle=al-Ris%C4%81la+al-muh%C4%ABt%C4%AByya%3A+A+Summary&rft.volume=22&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E85&rft.date=2010&rft_id=info%3Adoi%2F10.35834%2Fmjms%2F1312233136&rft.aulast=Azarian&rft.aufirst=Mohammad+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.35834%252Fmjms%252F1312233136&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" 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="CITEREFO'ConnorRobertson1999" class="citation web cs1">O'Connor, John J.; Robertson, Edmund F. (1999). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html">"Ghiyath al-Din Jamshid Mas'ud al-Kashi"</a>. <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110412192025/http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html">Archived</a> from the original on 12 April 2011. Retrieved 11 August 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor+History+of+Mathematics+archive&rft.atitle=Ghiyath+al-Din+Jamshid+Mas%27ud+al-Kashi&rft.date=1999&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2Fhistory%2FBiographies%2FAl-Kashi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006182-66">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006182_66-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006182_66-1">b</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006182_66-2">c</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 182. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006182–183-67"><a href="#cite_ref-FOOTNOTEArndtHaenel2006182–183_67-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 182–183. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006183-68"><a href="#cite_ref-FOOTNOTEArndtHaenel2006183_68-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 183. </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrienbergerus1630" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Christoph_Grienberger" title="Christoph Grienberger">Grienbergerus, Christophorus</a> (1630). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140201234124/http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf">Elementa Trigonometrica</a> (PDF) (in Latin). Archived from <a rel="nofollow" class="external text" href="https://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf">the original</a> (PDF) on 1 February 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementa+Trigonometrica&rft.date=1630&rft.aulast=Grienbergerus&rft.aufirst=Christophorus&rft_id=http%3A%2F%2Flibrarsi.comune.palermo.it%2Fgesuiti2%2F06.04.01.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < π < 3.14159 26535 89793 23846 26433 83279 50288 4199. </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="CITEREFBrezinski2009" class="citation book cs1">Brezinski, C. (2009). "Some pioneers of extrapolation methods". In <a href="/wiki/Adhemar_Bultheel" title="Adhemar Bultheel">Bultheel, Adhemar</a>; Cools, Ronald (eds.). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/doi/10.1142/9789812836267_0001">The Birth of Numerical Analysis</a>. World Scientific. pp. 1–22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789812836267_0001">10.1142/9789812836267_0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-283-625-0" title="Special:BookSources/978-981-283-625-0"><bdi>978-981-283-625-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Some+pioneers+of+extrapolation+methods&rft.btitle=The+Birth+of+Numerical+Analysis&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E22&rft.pub=World+Scientific&rft.date=2009&rft_id=info%3Adoi%2F10.1142%2F9789812836267_0001&rft.isbn=978-981-283-625-0&rft.aulast=Brezinski&rft.aufirst=C.&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fdoi%2F10.1142%2F9789812836267_0001&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoder1996" class="citation journal cs1"><a href="/wiki/Joella_Yoder" title="Joella Yoder">Yoder, Joella G.</a> (1996). <a rel="nofollow" class="external text" href="https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php">"Following in the footsteps of geometry: The mathematical world of Christiaan Huygens"</a>. De Zeventiende Eeuw. 12: 83–93 – via <a href="/wiki/Digital_Library_for_Dutch_Literature" title="Digital Library for Dutch Literature">Digital Library for Dutch Literature</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=De+Zeventiende+Eeuw&rft.atitle=Following+in+the+footsteps+of+geometry%3A+The+mathematical+world+of+Christiaan+Huygens&rft.volume=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E83-%3C%2Fspan%3E93&rft.date=1996&rft.aulast=Yoder&rft.aufirst=Joella+G.&rft_id=https%3A%2F%2Fwww.dbnl.org%2Ftekst%2F_zev001199601_01%2F_zev001199601_01_0009.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Ais-72"><a href="#cite_ref-Ais_72-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 185–191 </li> <li id="cite_note-Roypp-73">^ <a href="#cite_ref-Roypp_73-0">a</a> <a href="#cite_ref-Roypp_73-1">b</a> <a href="#cite_ref-Roypp_73-2">c</a> <a href="#cite_ref-Roypp_73-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRoy1990" class="citation journal cs1">Roy, Ranjan (1990). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230314224252/https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf">"The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha"</a> (PDF). Mathematics Magazine. 63 (5): 291–306. <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%2F0025570X.1990.11977541">10.1080/0025570X.1990.11977541</a>. Archived from <a rel="nofollow" class="external text" href="https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf">the original</a> (PDF) on 14 March 2023. Retrieved 21 February 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+Discovery+of+the+Series+Formula+for+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E+by+Leibniz%2C+Gregory+and+Nilakantha&rft.volume=63&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E291-%3C%2Fspan%3E306&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F0025570X.1990.11977541&rft.aulast=Roy&rft.aufirst=Ranjan&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fimages%2Fupload_library%2F22%2FAllendoerfer%2F1991%2F0025570x.di021167.02p0073q.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006185–186-74"><a href="#cite_ref-FOOTNOTEArndtHaenel2006185–186_74-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 185–186. </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph1991" class="citation book cs1">Joseph, George Gheverghese (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c-xT0KNJp0cC&pg=PA264">The Crest of the Peacock: Non-European Roots of Mathematics</a>. Princeton University Press. p. 264. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-13526-7" title="Special:BookSources/978-0-691-13526-7"><bdi>978-0-691-13526-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Crest+of+the+Peacock%3A+Non-European+Roots+of+Mathematics&rft.pages=264&rft.pub=Princeton+University+Press&rft.date=1991&rft.isbn=978-0-691-13526-7&rft.aulast=Joseph&rft.aufirst=George+Gheverghese&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc-xT0KNJp0cC%26pg%3DPA264&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006187-76">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006187_76-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006187_76-1">b</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 187. </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A060294" class="extiw" title="oeis:A060294">A060294</a> </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVieta1593" class="citation book cs1">Vieta, Franciscus (1593). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7_BCAAAAcAAJ">Variorum de rebus mathematicis responsorum</a>. 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Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica). 4: 75–83.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Universitatis+Scientiarum+Budapestiensis+%28Sectio+Computatorica%29&rft.atitle=On+the+Leibnizian+quadrature+of+the+circle.&rft.volume=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E75-%3C%2Fspan%3E83&rft.date=1983&rft.aulast=Horvath&rft.aufirst=Miklos&rft_id=http%3A%2F%2Fac.inf.elte.hu%2FVol_004_1983%2F075.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-LS-81">^ <a href="#cite_ref-LS_81-0">a</a> <a href="#cite_ref-LS_81-1">b</a> <a href="#CITEREFEymardLafon2004">Eymard & Lafon 2004</a>, pp. 53–54 </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCooker2011" class="citation journal cs1">Cooker, M.J. 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Archived from <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf">the original</a> (PDF) on 4 May 2019. Retrieved 23 February 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Gazette&rft.atitle=Fast+formulas+for+slowly+convergent+alternating+series&rft.volume=95&rft.issue=533&rft.pages=%3Cspan+class%3D%22nowrap%22%3E218-%3C%2Fspan%3E226&rft.date=2011&rft_id=info%3Adoi%2F10.1017%2FS0025557200002928&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123392772%23id-name%3DS2CID&rft.aulast=Cooker&rft.aufirst=M.J.&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fservices%2Faop-cambridge-core%2Fcontent%2Fview%2FF7C083868DEB95FE049CD44163367592%2FS0025557200002928a.pdf%2Fdiv-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006189-83"><a href="#cite_ref-FOOTNOTEArndtHaenel2006189_83-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 189. </li> <li id="cite_note-tweddle-84"><a href="#cite_ref-tweddle_84-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTweddle1991" class="citation journal cs1">Tweddle, Ian (1991). 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The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). Penguin. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-026149-3" title="Special:BookSources/978-0-14-026149-3"><bdi>978-0-14-026149-3</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+Penguin+Dictionary+of+Curious+and+Interesting+Numbers&rft.pages=35&rft.edition=revised&rft.pub=Penguin&rft.date=1997&rft.isbn=978-0-14-026149-3&rft.aulast=Wells&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Posamentier-98">^ <a href="#cite_ref-Posamentier_98-0">a</a> <a href="#cite_ref-Posamentier_98-1">b</a> <a href="#CITEREFPosamentierLehmann2004">Posamentier & Lehmann 2004</a>, p. 284 </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in <a href="#CITEREFBerggrenBorweinBorwein1997">Berggren, Borwein & Borwein 1997</a>, pp. 129–140 </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLindemann1882" class="citation journal cs1"><a href="/wiki/Ferdinand_Lindemann" class="mw-redirect" title="Ferdinand Lindemann">Lindemann, F.</a> (1882). <a rel="nofollow" class="external text" href="https://archive.org/details/sitzungsberichte1882deutsch/page/679">"Über die Ludolph'sche Zahl"</a>. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 2: 679–682.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=%C3%9Cber+die+Ludolph%27sche+Zahl&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E679-%3C%2Fspan%3E682&rft.date=1882&rft.aulast=Lindemann&rft.aufirst=F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsitzungsberichte1882deutsch%2Fpage%2F679&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006196-101"><a href="#cite_ref-FOOTNOTEArndtHaenel2006196_101-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 196. </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> Hardy and Wright 1938 and 2000: 177 footnote § 11.13–14 references Lindemann's proof as appearing at Math. Ann. 20 (1882), 213–225. </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> cf Hardy and Wright 1938 and 2000:177 footnote § 11.13–14. The proofs that e and π are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full citations. </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOughtred1648" class="citation book cs1 cs1-prop-foreign-lang-source">Oughtred, William (1648). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_ddMxgr27tNkC">Clavis Mathematicæ</a> [The key to mathematics] (in Latin). London: Thomas Harper. p. <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_ddMxgr27tNkC/page/n220">69</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Clavis+Mathematic%C3%A6&rft.place=London&rft.pages=69&rft.pub=Thomas+Harper&rft.date=1648&rft.aulast=Oughtred&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_ddMxgr27tNkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> (English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOughtred1694" class="citation book cs1">Oughtred, William (1694). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S50yAQAAMAAJ&pg=PA99">Key of the Mathematics</a>. J. Salusbury.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Key+of+the+Mathematics&rft.pub=J.+Salusbury&rft.date=1694&rft.aulast=Oughtred&rft.aufirst=William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DS50yAQAAMAAJ%26pg%3DPA99&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988">) </li> <li id="cite_note-FOOTNOTEArndtHaenel2006166-105">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006166_105-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006166_105-1">b</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006166_105-2">c</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006166_105-3">d</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 166. </li> <li id="cite_note-Cajori-2007-106">^ <a href="#cite_ref-Cajori-2007_106-0">a</a> <a href="#cite_ref-Cajori-2007_106-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCajori2007" class="citation book cs1">Cajori, Florian (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bT5suOONXlgC&pg=PA9">A History of Mathematical Notations: Vol. II</a>. Cosimo, Inc. pp. 8–13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60206-714-1" title="Special:BookSources/978-1-60206-714-1"><bdi>978-1-60206-714-1</bdi></a>. <q>the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented 3.14159... by δ:π, as did Oughtred more than a century earlier</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Notations%3A+Vol.+II&rft.pages=%3Cspan+class%3D%22nowrap%22%3E8-%3C%2Fspan%3E13&rft.pub=Cosimo%2C+Inc.&rft.date=2007&rft.isbn=978-1-60206-714-1&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbT5suOONXlgC%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Smith-1958-107">^ <a href="#cite_ref-Smith-1958_107-0">a</a> <a href="#cite_ref-Smith-1958_107-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1958" class="citation book cs1">Smith, David E. (1958). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC&pg=PA312">History of Mathematics</a>. Courier Corporation. p. 312. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-20430-7" title="Special:BookSources/978-0-486-20430-7"><bdi>978-0-486-20430-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pages=312&rft.pub=Courier+Corporation&rft.date=1958&rft.isbn=978-0-486-20430-7&rft.aulast=Smith&rft.aufirst=David+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuTytJGnTf1kC%26pg%3DPA312&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArchibald1921" class="citation journal cs1">Archibald, R.C. (1921). "Historical Notes on the Relation e−(π/2) = ii". The American Mathematical Monthly. 28 (3): 116–121. <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%2F2972388">10.2307/2972388</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/2972388">2972388</a>. <q>It is noticeable that these letters are never used separately, that is, π is not used for 'Semiperipheria'</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Historical+Notes+on+the+Relation+%3Cspan+class%3D%22texhtml+%22+%3Ee%3Csup%3E%E2%88%92%28%CF%80%2F2%29%3C%2Fsup%3E+%3D+i%3Csup%3Ei%3C%2Fsup%3E%3C%2Fspan%3E&rft.volume=28&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E116-%3C%2Fspan%3E121&rft.date=1921&rft_id=info%3Adoi%2F10.2307%2F2972388&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2972388%23id-name%3DJSTOR&rft.aulast=Archibald&rft.aufirst=R.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarrow1860" class="citation book cs1 cs1-prop-foreign-lang-source">Barrow, Isaac (1860). <a rel="nofollow" class="external text" href="https://archive.org/stream/mathematicalwor00whewgoog#page/n405/mode/1up">"Lecture XXIV"</a>. In Whewell, William (ed.). The mathematical works of Isaac Barrow (in Latin). Harvard University. Cambridge University press. p. 381.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lecture+XXIV&rft.btitle=The+mathematical+works+of+Isaac+Barrow&rft.pages=381&rft.pub=Cambridge+University+press&rft.date=1860&rft.aulast=Barrow&rft.aufirst=Isaac&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fmathematicalwor00whewgoog%23page%2Fn405%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGregorius1695" class="citation journal cs1 cs1-prop-foreign-lang-source">Gregorius, David (1695). <a rel="nofollow" class="external text" href="https://archive.org/download/crossref-pre-1909-scholarly-works/10.1098%252Frstl.1684.0084.zip/10.1098%252Frstl.1695.0114.pdf">"Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae"</a> (PDF). Philosophical Transactions (in Latin). 19 (231): 637–652. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1695RSPT...19..637G">1695RSPT...19..637G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1695.0114">10.1098/rstl.1695.0114</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/102382">102382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions&rft.atitle=Ad+Reverendum+Virum+D.+Henricum+Aldrich+S.T.T.+Decanum+Aedis+Christi+Oxoniae&rft.volume=19&rft.issue=231&rft.pages=%3Cspan+class%3D%22nowrap%22%3E637-%3C%2Fspan%3E652&rft.date=1695&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F102382%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frstl.1695.0114&rft_id=info%3Abibcode%2F1695RSPT...19..637G&rft.aulast=Gregorius&rft.aufirst=David&rft_id=https%3A%2F%2Farchive.org%2Fdownload%2Fcrossref-pre-1909-scholarly-works%2F10.1098%25252Frstl.1684.0084.zip%2F10.1098%25252Frstl.1695.0114.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006165-111"><a href="#cite_ref-FOOTNOTEArndtHaenel2006165_111-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 165: A facsimile of Jones' text is in <a href="#CITEREFBerggrenBorweinBorwein1997">Berggren, Borwein & Borwein 1997</a>, pp. 108–109. </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSegner1756" class="citation book cs1 cs1-prop-foreign-lang-source">Segner, Joannes Andreas (1756). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NmYVAAAAQAAJ&pg=PA282">Cursus Mathematicus</a> (in Latin). Halae Magdeburgicae. p. 282. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171015150340/https://books.google.co.uk/books?id=NmYVAAAAQAAJ&pg=PA282">Archived</a> from the original on 15 October 2017. Retrieved 15 October 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cursus+Mathematicus&rft.pages=282&rft.pub=Halae+Magdeburgicae&rft.date=1756&rft.aulast=Segner&rft.aufirst=Joannes+Andreas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNmYVAAAAQAAJ%26pg%3DPA282&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuler1727" class="citation journal cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1727). <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5">"Tentamen explicationis phaenomenorum aeris"</a> (PDF). Commentarii Academiae Scientiarum Imperialis Petropolitana (in Latin). 2: 351. <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/pages/E007.html">E007</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160401072718/http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5">Archived</a> (PDF) from the original on 1 April 2016. Retrieved 15 October 2017. <q>Sumatur pro ratione radii ad peripheriem, I : π</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commentarii+Academiae+Scientiarum+Imperialis+Petropolitana&rft.atitle=Tentamen+explicationis+phaenomenorum+aeris&rft.volume=2&rft.pages=351&rft.date=1727&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=http%3A%2F%2Feulerarchive.maa.org%2Fdocs%2Foriginals%2FE007.pdf%23page%3D5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> <a rel="nofollow" class="external text" href="http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3">English translation by Ian Bruce</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160610172054/http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3">Archived</a> 10 June 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: "π is taken for the ratio of the radius to the periphery [note that in this work, Euler's π is double our π.]" </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuler1747" class="citation book cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1747). Henry, Charles (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3C1iHFBXVEcC&pg=PA139">Lettres inédites d'Euler à d'Alembert</a>. Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche (in French). Vol. 19 (published 1886). p. 139. <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/pages/E858.html">E858</a>. <q>Car, soit π la circonference d'un cercle, dout le rayon est = 1</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lettres+in%C3%A9dites+d%27Euler+%C3%A0+d%27Alembert&rft.series=Bullettino+di+Bibliografia+e+di+Storia+delle+Scienze+Matematiche+e+Fisiche&rft.pages=139&rft.date=1747&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3C1iHFBXVEcC%26pg%3DPA139&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> English translation in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCajori1913" class="citation journal cs1">Cajori, Florian (1913). "History of the Exponential and Logarithmic Concepts". The American Mathematical Monthly. 20 (3): 75–84. <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%2F2973441">10.2307/2973441</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/2973441">2973441</a>. <q>Letting π be the circumference (!) of a circle of unit radius</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=History+of+the+Exponential+and+Logarithmic+Concepts&rft.volume=20&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E75-%3C%2Fspan%3E84&rft.date=1913&rft_id=info%3Adoi%2F10.2307%2F2973441&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2973441%23id-name%3DJSTOR&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuler1736" class="citation book cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1736). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jgdTAAAAcAAJ&pg=PA113">"Ch. 3 Prop. 34 Cor. 1"</a>. Mechanica sive motus scientia analytice exposita. (cum tabulis) (in Latin). Vol. 1. Academiae scientiarum Petropoli. p. 113. <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/pages/E015.html">E015</a>. <q>Denotet 1 : π rationem diametri ad peripheriam</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ch.+3+Prop.+34+Cor.+1&rft.btitle=Mechanica+sive+motus+scientia+analytice+exposita.+%28cum+tabulis%29&rft.pages=113&rft.pub=Academiae+scientiarum+Petropoli&rft.date=1736&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjgdTAAAAcAAJ%26pg%3DPA113&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> <a rel="nofollow" class="external text" href="http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26">English translation by Ian Bruce</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160610183753/http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26">Archived</a> 10 June 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> : "Let 1 : π denote the ratio of the diameter to the circumference" </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuler1922" class="citation book cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1922). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155">Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio</a> (in Latin). Lipsae: B.G. Teubneri. pp. 133–134. <a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/pages/E101.html">E101</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171016022758/http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155">Archived</a> from the original on 16 October 2017. Retrieved 15 October 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Leonhardi+Euleri+opera+omnia.+1%2C+Opera+mathematica.+Volumen+VIII%2C+Leonhardi+Euleri+introductio+in+analysin+infinitorum.+Tomus+primus+%2F+ediderunt+Adolf+Krazer+et+Ferdinand+Rudio&rft.place=Lipsae&rft.pages=%3Cspan+class%3D%22nowrap%22%3E133-%3C%2Fspan%3E134&rft.pub=B.G.+Teubneri&rft.date=1922&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k69587%2Ff155&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSegner1761" class="citation book cs1 cs1-prop-foreign-lang-source">Segner, Johann Andreas von (1761). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P-hEAAAAcAAJ&pg=PA374">Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm</a> (in Latin). Renger. p. 374. <q>Si autem π notet peripheriam circuli, cuius diameter eſt 2</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cursus+Mathematicus%3A+Elementorum+Analyseos+Infinitorum+Elementorum+Analyseos+Infinitorvm&rft.pages=374&rft.pub=Renger&rft.date=1761&rft.aulast=Segner&rft.aufirst=Johann+Andreas+von&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP-hEAAAAcAAJ%26pg%3DPA374&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel2006205-118"><a href="#cite_ref-FOOTNOTEArndtHaenel2006205_118-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 205. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006197-119">^ <a href="#cite_ref-FOOTNOTEArndtHaenel2006197_119-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel2006197_119-1">b</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 197. </li> <li id="cite_note-120"><a href="#cite_ref-120">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReitwiesner1950" class="citation journal cs1">Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places". Mathematical Tables and Other Aids to Computation. 4 (29): 11–15. <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%2F2002695">10.2307/2002695</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/2002695">2002695</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Tables+and+Other+Aids+to+Computation&rft.atitle=An+ENIAC+Determination+of+pi+and+e+to+2000+Decimal+Places&rft.volume=4&rft.issue=29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E11-%3C%2Fspan%3E15&rft.date=1950&rft_id=info%3Adoi%2F10.2307%2F2002695&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2002695%23id-name%3DJSTOR&rft.aulast=Reitwiesner&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNicholsonJeenel1955" class="citation journal cs1">Nicholson, J. C.; Jeenel, J. (1955). "Some comments on a NORC Computation of π". Math. Tabl. Aids. Comp. 9 (52): 162–164. <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%2F2002052">10.2307/2002052</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/2002052">2002052</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Tabl.+Aids.+Comp.&rft.atitle=Some+comments+on+a+NORC+Computation+of+%CF%80&rft.volume=9&rft.issue=52&rft.pages=%3Cspan+class%3D%22nowrap%22%3E162-%3C%2Fspan%3E164&rft.date=1955&rft_id=info%3Adoi%2F10.2307%2F2002052&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2002052%23id-name%3DJSTOR&rft.aulast=Nicholson&rft.aufirst=J.+C.&rft.au=Jeenel%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel200615–17-122"><a href="#cite_ref-FOOTNOTEArndtHaenel200615–17_122-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 15–17. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006131-123"><a href="#cite_ref-FOOTNOTEArndtHaenel2006131_123-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 131. </li> <li id="cite_note-FOOTNOTEArndtHaenel2006132,_140-124"><a href="#cite_ref-FOOTNOTEArndtHaenel2006132,_140_124-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 132, 140. </li> <li id="cite_note-FOOTNOTEArndtHaenel200687-125">^ <a href="#cite_ref-FOOTNOTEArndtHaenel200687_125-0">a</a> <a href="#cite_ref-FOOTNOTEArndtHaenel200687_125-1">b</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 87. </li> <li id="cite_note-126"><a href="#cite_ref-126">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 111 (5 times), pp. 113–114 (4 times). For details of algorithms, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBorweinBorwein1987" class="citation book cs1">Borwein, Jonathan; Borwein, Peter (1987). Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-31515-5" title="Special:BookSources/978-0-471-31515-5"><bdi>978-0-471-31515-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+a+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.pub=Wiley&rft.date=1987&rft.isbn=978-0-471-31515-5&rft.aulast=Borwein&rft.aufirst=Jonathan&rft.au=Borwein%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Background-127">^ <a href="#cite_ref-Background_127-0">a</a> <a href="#cite_ref-Background_127-1">b</a> <a href="#cite_ref-Background_127-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBailey2003" class="citation web cs1">Bailey, David H. (16 May 2003). <a rel="nofollow" class="external text" href="http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/dhb-kanada.pdf">"Some Background on Kanada's Recent Pi Calculation"</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120415102439/http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/dhb-kanada.pdf">Archived</a> (PDF) from the original on 15 April 2012. Retrieved 12 April 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Some+Background+on+Kanada%27s+Recent+Pi+Calculation&rft.date=2003-05-16&rft.aulast=Bailey&rft.aufirst=David+H.&rft_id=http%3A%2F%2Fcrd-legacy.lbl.gov%2F~dhbailey%2Fdhbpapers%2Fdhb-kanada.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-128"><a href="#cite_ref-128">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 17–19 </li> <li id="cite_note-MSNBC-129"><a href="#cite_ref-MSNBC_129-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchudel2009" class="citation news cs1">Schudel, Matt (25 March 2009). "John W. Wrench, Jr.: Mathematician Had a Taste for Pi". The Washington Post. p. B5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Washington+Post&rft.atitle=John+W.+Wrench%2C+Jr.%3A+Mathematician+Had+a+Taste+for+Pi&rft.pages=B5&rft.date=2009-03-25&rft.aulast=Schudel&rft.aufirst=Matt&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-independent.co.uk-130"><a href="#cite_ref-independent.co.uk_130-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConnor2010" class="citation news cs1">Connor, Steve (8 January 2010). <a rel="nofollow" class="external text" href="https://www.independent.co.uk/news/science/the-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html">"The Big Question: How close have we come to knowing the precise value of pi?"</a>. The Independent. London. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120402220803/http://www.independent.co.uk/news/science/the-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html">Archived</a> from the original on 2 April 2012. Retrieved 14 April 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Independent&rft.atitle=The+Big+Question%3A+How+close+have+we+come+to+knowing+the+precise+value+of+pi%3F&rft.date=2010-01-08&rft.aulast=Connor&rft.aufirst=Steve&rft_id=https%3A%2F%2Fwww.independent.co.uk%2Fnews%2Fscience%2Fthe-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArndtHaenel200618-131"><a href="#cite_ref-FOOTNOTEArndtHaenel200618_131-0">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 18. </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 103–104 </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 104 </li> <li id="cite_note-134"><a href="#cite_ref-134">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 104, 206 </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 110–111 </li> <li id="cite_note-136"><a href="#cite_ref-136">^</a> <a href="#CITEREFEymardLafon2004">Eymard & Lafon 2004</a>, p. 254 </li> <li id="cite_note-NW-137">^ <a href="#cite_ref-NW_137-0">a</a> <a href="#cite_ref-NW_137-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaileyBorwein2016" class="citation book cs1"><a href="/wiki/David_H._Bailey_(mathematician)" title="David H. Bailey (mathematician)">Bailey, David H.</a>; <a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan M.</a> (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K26zDAAAQBAJ&pg=PA469">"15.2 Computational records"</a>. Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation. Springer International Publishing. p. 469. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-32377-0">10.1007/978-3-319-32377-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-32375-6" title="Special:BookSources/978-3-319-32375-6"><bdi>978-3-319-32375-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=15.2+Computational+records&rft.btitle=Pi%3A+The+Next+Generation%2C+A+Sourcebook+on+the+Recent+History+of+Pi+and+Its+Computation&rft.pages=469&rft.pub=Springer+International+Publishing&rft.date=2016&rft_id=info%3Adoi%2F10.1007%2F978-3-319-32377-0&rft.isbn=978-3-319-32375-6&rft.aulast=Bailey&rft.aufirst=David+H.&rft.au=Borwein%2C+Jonathan+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK26zDAAAQBAJ%26pg%3DPA469&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-138"><a href="#cite_ref-138">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCassel2022" class="citation magazine cs1">Cassel, David (11 June 2022). <a rel="nofollow" class="external text" href="https://thenewstack.io/how-googles-emma-haruka-iwao-helped-set-a-new-record-for-pi/">"How Google's Emma Haruka Iwao Helped Set a New Record for Pi"</a>. The New Stack.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+Stack&rft.atitle=How+Google%27s+Emma+Haruka+Iwao+Helped+Set+a+New+Record+for+Pi&rft.date=2022-06-11&rft.aulast=Cassel&rft.aufirst=David&rft_id=https%3A%2F%2Fthenewstack.io%2Fhow-googles-emma-haruka-iwao-helped-set-a-new-record-for-pi%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> PSLQ means Partial Sum of Least Squares. </li> <li id="cite_note-140"><a href="#cite_ref-140">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPlouffe2006" class="citation web cs1"><a href="/wiki/Simon_Plouffe" title="Simon Plouffe">Plouffe, Simon</a> (April 2006). <a rel="nofollow" class="external text" href="http://plouffe.fr/simon/inspired2.pdf">"Identities inspired by Ramanujan's Notebooks (part 2)"</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120114101641/http://www.plouffe.fr/simon/inspired2.pdf">Archived</a> (PDF) from the original on 14 January 2012. Retrieved 10 April 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Identities+inspired+by+Ramanujan%27s+Notebooks+%28part+2%29&rft.date=2006-04&rft.aulast=Plouffe&rft.aufirst=Simon&rft_id=http%3A%2F%2Fplouffe.fr%2Fsimon%2Finspired2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-141"><a href="#cite_ref-141">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 39 </li> <li id="cite_note-bn-142"><a href="#cite_ref-bn_142-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRamaley1969" class="citation journal cs1">Ramaley, J.F. (October 1969). "Buffon's Needle Problem". The American Mathematical Monthly. 76 (8): 916–918. <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%2F2317945">10.2307/2317945</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/2317945">2317945</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+Mathematical+Monthly&rft.atitle=Buffon%27s+Needle+Problem&rft.volume=76&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E916-%3C%2Fspan%3E918&rft.date=1969-10&rft_id=info%3Adoi%2F10.2307%2F2317945&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2317945%23id-name%3DJSTOR&rft.aulast=Ramaley&rft.aufirst=J.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-143"><a href="#cite_ref-143">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 39–40 <a href="#CITEREFPosamentierLehmann2004">Posamentier & Lehmann 2004</a>, p. 105 </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrünbaum1960" class="citation journal cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, B.</a> (1960). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1960-0114110-9">"Projection Constants"</a>. <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 95 (3): 451–465. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1960-0114110-9">10.1090/s0002-9947-1960-0114110-9</a>.</cite><span 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Retrieved 22 January 2025. <q>A=1/2(2πr)r=πr^2</q></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=Circle&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCircle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> <a href="#CITEREFBronshteĭnSemendiaev1971">Bronshteĭn & Semendiaev 1971</a>, pp. 200, 209 </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Circumference.html">"Circumference"</a>. mathworld.wolfram.com. Retrieved 22 January 2025. <q>C=2πr</q></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=Circumference&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCircumference.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-158"><a href="#cite_ref-158">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Ellipse.html">"Ellipse"</a>. mathworld.wolfram.com. Retrieved 22 January 2025. <q>A=...πab.</q></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=Ellipse&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FEllipse.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-159"><a href="#cite_ref-159">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMartiniMontejanoOliveros2019" class="citation book cs1">Martini, Horst; Montejano, Luis; <a href="/wiki/D%C3%A9borah_Oliveros" title="Déborah Oliveros">Oliveros, Déborah</a> (2019). Bodies of Constant Width: An Introduction to Convex Geometry with Applications. Birkhäuser. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-03868-7">10.1007/978-3-030-03868-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-03866-3" title="Special:BookSources/978-3-030-03866-3"><bdi>978-3-030-03866-3</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=3930585">3930585</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:127264210">127264210</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bodies+of+Constant+Width%3A+An+Introduction+to+Convex+Geometry+with+Applications&rft.pub=Birkh%C3%A4user&rft.date=2019&rft_id=info%3Adoi%2F10.1007%2F978-3-030-03868-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3930585%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A127264210%23id-name%3DS2CID&rft.isbn=978-3-030-03866-3&rft.aulast=Martini&rft.aufirst=Horst&rft.au=Montejano%2C+Luis&rft.au=Oliveros%2C+D%C3%A9borah&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"><div class="paragraphbreak" style="margin-top:0.5em"></div> See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, § 5.3.3, pp. 111–112. </li> <li id="cite_note-160"><a href="#cite_ref-160">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHermanStrang2016" class="citation book cs1">Herman, Edwin; <a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2016). <a rel="nofollow" class="external text" href="https://openstax.org/books/calculus-volume-1/pages/5-5-substitution">"Section 5.5, Exercise 316"</a>. 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Publish or Perish Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Comprehensive+Introduction+to+Differential+Geometry&rft.pub=Publish+or+Perish+Press&rft.date=1999&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988">; Chapter 6. </li> <li id="cite_note-179"><a href="#cite_ref-179">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKobayashiNomizu1996" class="citation book cs1">Kobayashi, Shoshichi; Nomizu, Katsumi (1996). <a href="/wiki/Foundations_of_Differential_Geometry" title="Foundations of Differential Geometry">Foundations of Differential Geometry</a>. Vol. 2 (New ed.). <a href="/wiki/Wiley_Interscience" class="mw-redirect" title="Wiley Interscience">Wiley Interscience</a>. p. 293.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Differential+Geometry&rft.pages=293&rft.edition=New&rft.pub=Wiley+Interscience&rft.date=1996&rft.aulast=Kobayashi&rft.aufirst=Shoshichi&rft.au=Nomizu%2C+Katsumi&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988">; Chapter XII Characteristic classes </li> <li id="cite_note-180"><a href="#cite_ref-180">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAhlfors1966" class="citation book cs1"><a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Ahlfors, Lars</a> (1966). Complex analysis. McGraw-Hill. p. 115.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.pages=115&rft.pub=McGraw-Hill&rft.date=1966&rft.aulast=Ahlfors&rft.aufirst=Lars&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-181"><a href="#cite_ref-181">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoglekar2005" class="citation book cs1">Joglekar, S. D. (2005). Mathematical Physics. Universities Press. p. 166. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7371-422-1" title="Special:BookSources/978-81-7371-422-1"><bdi>978-81-7371-422-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Physics&rft.pages=166&rft.pub=Universities+Press&rft.date=2005&rft.isbn=978-81-7371-422-1&rft.aulast=Joglekar&rft.aufirst=S.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-182"><a href="#cite_ref-182">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchey1996" class="citation book cs1">Schey, H. M. (1996). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. W.W. Norton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-96997-5" title="Special:BookSources/0-393-96997-5"><bdi>0-393-96997-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Div%2C+Grad%2C+Curl%2C+and+All+That%3A+An+Informal+Text+on+Vector+Calculus&rft.pub=W.W.+Norton&rft.date=1996&rft.isbn=0-393-96997-5&rft.aulast=Schey&rft.aufirst=H.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-183"><a href="#cite_ref-183">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYeo2006" class="citation book cs1">Yeo, Adrian (2006). The pleasures of pi, e and other interesting numbers. World Scientific Pub. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-270-078-0" title="Special:BookSources/978-981-270-078-0"><bdi>978-981-270-078-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+pleasures+of+pi%2C+e+and+other+interesting+numbers&rft.pages=21&rft.pub=World+Scientific+Pub.&rft.date=2006&rft.isbn=978-981-270-078-0&rft.aulast=Yeo&rft.aufirst=Adrian&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-184"><a href="#cite_ref-184">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEhlers2000" class="citation book cs1">Ehlers, Jürgen (2000). Einstein's Field Equations and Their Physical Implications. Springer. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-67073-5" title="Special:BookSources/978-3-540-67073-5"><bdi>978-3-540-67073-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+Field+Equations+and+Their+Physical+Implications&rft.pages=7&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-540-67073-5&rft.aulast=Ehlers&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-Elliptic_PDE2-185">^ <a href="#cite_ref-Elliptic_PDE2_185-0">a</a> <a href="#cite_ref-Elliptic_PDE2_185-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGilbargTrudinger1983" class="citation cs2">Gilbarg, D.; <a href="/wiki/Neil_Trudinger" title="Neil Trudinger">Trudinger, Neil</a> (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-41160-7" title="Special:BookSources/3-540-41160-7"><bdi>3-540-41160-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Partial+Differential+Equations+of+Second+Order&rft.place=New+York&rft.pub=Springer&rft.date=1983&rft.isbn=3-540-41160-7&rft.aulast=Gilbarg&rft.aufirst=D.&rft.au=Trudinger%2C+Neil&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-186"><a href="#cite_ref-186">^</a> <a href="#CITEREFBronshteĭnSemendiaev1971">Bronshteĭn & Semendiaev 1971</a>, pp. 191–192 </li> <li id="cite_note-187"><a href="#cite_ref-187">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArtin1964" class="citation book cs1"><a href="/wiki/Emil_Artin" title="Emil Artin">Artin, Emil</a> (1964). The Gamma Function. Athena series; selected topics in mathematics (1st ed.). Holt, Rinehart and Winston.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Gamma+Function&rft.series=Athena+series%3B+selected+topics+in+mathematics&rft.edition=1st&rft.pub=Holt%2C+Rinehart+and+Winston&rft.date=1964&rft.aulast=Artin&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-188"><a href="#cite_ref-188">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEvans1997" class="citation book cs1">Evans, Lawrence (1997). Partial Differential Equations. AMS. p. 615.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations&rft.pages=615&rft.pub=AMS&rft.date=1997&rft.aulast=Evans&rft.aufirst=Lawrence&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-189"><a href="#cite_ref-189">^</a> <a href="#CITEREFBronshteĭnSemendiaev1971">Bronshteĭn & Semendiaev 1971</a>, p. 190 </li> <li id="cite_note-190"><a href="#cite_ref-190">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBenjamin_NillAndreas_Paffenholz2014" class="citation journal cs1">Benjamin Nill; Andreas Paffenholz (2014). "On the equality case in Erhart's volume conjecture". Advances in Geometry. 14 (4): 579–586. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1205.1270">1205.1270</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fadvgeom-2014-0001">10.1515/advgeom-2014-0001</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/1615-7168">1615-7168</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:119125713">119125713</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Geometry&rft.atitle=On+the+equality+case+in+Erhart%27s+volume+conjecture&rft.volume=14&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E579-%3C%2Fspan%3E586&rft.date=2014&rft_id=info%3Aarxiv%2F1205.1270&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119125713%23id-name%3DS2CID&rft.issn=1615-7168&rft_id=info%3Adoi%2F10.1515%2Fadvgeom-2014-0001&rft.au=Benjamin+Nill&rft.au=Andreas+Paffenholz&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-191"><a href="#cite_ref-191">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 41–43 </li> <li id="cite_note-192"><a href="#cite_ref-192">^</a> This theorem was proved by <a href="/wiki/Ernesto_Ces%C3%A0ro" title="Ernesto Cesàro">Ernesto Cesàro</a> in 1881. For a more rigorous proof than the intuitive and informal one given here, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHardy2008" class="citation book cs1">Hardy, G. H. (2008). An Introduction to the Theory of Numbers. Oxford University Press. Theorem 332. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921986-5" title="Special:BookSources/978-0-19-921986-5"><bdi>978-0-19-921986-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.pages=Theorem+332&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-921986-5&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-193"><a href="#cite_ref-193">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOgilvyAnderson1988" class="citation book cs1"><a href="/wiki/C._Stanley_Ogilvy" title="C. Stanley Ogilvy">Ogilvy, C. S.</a>; Anderson, J. T. (1988). Excursions in Number Theory. Dover Publications Inc. pp. 29–35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-25778-9" title="Special:BookSources/0-486-25778-9"><bdi>0-486-25778-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Excursions+in+Number+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E29-%3C%2Fspan%3E35&rft.pub=Dover+Publications+Inc.&rft.date=1988&rft.isbn=0-486-25778-9&rft.aulast=Ogilvy&rft.aufirst=C.+S.&rft.au=Anderson%2C+J.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-194"><a href="#cite_ref-194">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 43 </li> <li id="cite_note-195"><a href="#cite_ref-195">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPlatonovRapinchuk1994" class="citation book cs1">Platonov, Vladimir; Rapinchuk, Andrei (1994). Algebraic Groups and Number Theory. Academic Press. pp. 262–265.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Groups+and+Number+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E262-%3C%2Fspan%3E265&rft.pub=Academic+Press&rft.date=1994&rft.aulast=Platonov&rft.aufirst=Vladimir&rft.au=Rapinchuk%2C+Andrei&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-196"><a href="#cite_ref-196">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSondow1994" class="citation journal cs1">Sondow, J. (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series". <a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a>. 120 (2): 421–424. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.352.5774">10.1.1.352.5774</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9939-1994-1172954-7">10.1090/s0002-9939-1994-1172954-7</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:122276856">122276856</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Analytic+continuation+of+Riemann%27s+zeta+function+and+values+at+negative+integers+via+Euler%27s+transformation+of+series&rft.volume=120&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E421-%3C%2Fspan%3E424&rft.date=1994&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.352.5774%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122276856%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2Fs0002-9939-1994-1172954-7&rft.aulast=Sondow&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-197"><a href="#cite_ref-197">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFT._FriedmannC.R._Hagen2015" class="citation journal cs1">T. Friedmann; C.R. Hagen (2015). "Quantum mechanical derivation of the Wallis formula for pi". Journal of Mathematical Physics. 56 (11): 112101. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1510.07813">1510.07813</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JMP....56k2101F">2015JMP....56k2101F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4930800">10.1063/1.4930800</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:119315853">119315853</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Quantum+mechanical+derivation+of+the+Wallis+formula+for+pi&rft.volume=56&rft.issue=11&rft.pages=112101&rft.date=2015&rft_id=info%3Aarxiv%2F1510.07813&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119315853%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.4930800&rft_id=info%3Abibcode%2F2015JMP....56k2101F&rft.au=T.+Friedmann&rft.au=C.R.+Hagen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-198"><a href="#cite_ref-198">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTate1950" class="citation conference cs1">Tate, John T. (1950). "Fourier analysis in number fields, and Hecke's zeta-functions". In Cassels, J. W. S.; Fröhlich, A. (eds.). Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, DC. pp. 305–347. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9502734-2-6" title="Special:BookSources/978-0-9502734-2-6"><bdi>978-0-9502734-2-6</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=0217026">0217026</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Fourier+analysis+in+number+fields%2C+and+Hecke%27s+zeta-functions&rft.btitle=Algebraic+Number+Theory+%28Proc.+Instructional+Conf.%2C+Brighton%2C+1965%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E305-%3C%2Fspan%3E347&rft.pub=Thompson%2C+Washington%2C+DC&rft.date=1950&rft.isbn=978-0-9502734-2-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0217026%23id-name%3DMR&rft.aulast=Tate&rft.aufirst=John+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDymMcKean1972Chapter_4-199"><a href="#cite_ref-FOOTNOTEDymMcKean1972Chapter_4_199-0">^</a> <a href="#CITEREFDymMcKean1972">Dym & McKean 1972</a>, Chapter 4. </li> <li id="cite_note-Mumford_1983_1–117-200">^ <a href="#cite_ref-Mumford_1983_1–117_200-0">a</a> <a href="#cite_ref-Mumford_1983_1–117_200-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMumford1983" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1983). Tata Lectures on Theta I. Boston: Birkhauser. pp. 1–117. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-3109-2" title="Special:BookSources/978-3-7643-3109-2"><bdi>978-3-7643-3109-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tata+Lectures+on+Theta+I&rft.place=Boston&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E117&rft.pub=Birkhauser&rft.date=1983&rft.isbn=978-3-7643-3109-2&rft.aulast=Mumford&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-201"><a href="#cite_ref-201">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPortStone1978" class="citation book cs1">Port, Sidney; Stone, Charles (1978). Brownian motion and classical potential theory. Academic Press. p. 29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Brownian+motion+and+classical+potential+theory&rft.pages=29&rft.pub=Academic+Press&rft.date=1978&rft.aulast=Port&rft.aufirst=Sidney&rft.au=Stone%2C+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-202"><a href="#cite_ref-202">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTitchmarsh1948" class="citation book cs1"><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, E.</a> (1948). Introduction to the Theory of Fourier Integrals (2nd ed.). Oxford University: Clarendon Press (published 1986). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-0324-5" title="Special:BookSources/978-0-8284-0324-5"><bdi>978-0-8284-0324-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Fourier+Integrals&rft.place=Oxford+University&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1948&rft.isbn=978-0-8284-0324-5&rft.aulast=Titchmarsh&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-203"><a href="#cite_ref-203">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStein1970" class="citation book cs1"><a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a> (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Singular+Integrals+and+Differentiability+Properties+of+Functions&rft.pub=Princeton+University+Press&rft.date=1970&rft.aulast=Stein&rft.aufirst=Elias&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988">; Chapter II. </li> <li id="cite_note-KA-204">^ <a href="#cite_ref-KA_204-0">a</a> <a href="#cite_ref-KA_204-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKlebanoff2001" class="citation journal cs1">Klebanoff, Aaron (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111027155739/http://home.comcast.net/~davejanelle/mandel.pdf">"Pi in the Mandelbrot set"</a> (PDF). Fractals. 9 (4): 393–402. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218348X01000828">10.1142/S0218348X01000828</a>. Archived from <a rel="nofollow" class="external text" href="http://home.comcast.net/~davejanelle/mandel.pdf">the original</a> (PDF) on 27 October 2011. Retrieved 14 April 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fractals&rft.atitle=Pi+in+the+Mandelbrot+set&rft.volume=9&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E393-%3C%2Fspan%3E402&rft.date=2001&rft_id=info%3Adoi%2F10.1142%2FS0218348X01000828&rft.aulast=Klebanoff&rft.aufirst=Aaron&rft_id=http%3A%2F%2Fhome.comcast.net%2F~davejanelle%2Fmandel.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-205"><a href="#cite_ref-205">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeitgen2004" class="citation book cs1">Peitgen, Heinz-Otto (2004). Chaos and fractals: new frontiers of science. Springer. pp. 801–803. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20229-7" title="Special:BookSources/978-0-387-20229-7"><bdi>978-0-387-20229-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos+and+fractals%3A+new+frontiers+of+science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E801-%3C%2Fspan%3E803&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-20229-7&rft.aulast=Peitgen&rft.aufirst=Heinz-Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-206"><a href="#cite_ref-206">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOvsienkoTabachnikov2004" class="citation book cs1">Ovsienko, V.; Tabachnikov, S. (2004). "Section 1.3". Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge Tracts in Mathematics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83186-4" title="Special:BookSources/978-0-521-83186-4"><bdi>978-0-521-83186-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+1.3&rft.btitle=Projective+Differential+Geometry+Old+and+New%3A+From+the+Schwarzian+Derivative+to+the+Cohomology+of+Diffeomorphism+Groups&rft.series=Cambridge+Tracts+in+Mathematics&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-83186-4&rft.aulast=Ovsienko&rft.aufirst=V.&rft.au=Tabachnikov%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-207"><a href="#cite_ref-207">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHallidayResnickWalker1997" class="citation book cs1">Halliday, David; Resnick, Robert; Walker, Jearl (1997). Fundamentals of Physics (5th ed.). John Wiley & Sons. p. 381. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-14854-7" title="Special:BookSources/0-471-14854-7"><bdi>0-471-14854-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Physics&rft.pages=381&rft.edition=5th&rft.pub=John+Wiley+%26+Sons&rft.date=1997&rft.isbn=0-471-14854-7&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rft.au=Walker%2C+Jearl&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-208"><a href="#cite_ref-208">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFUroneHinrichs2022" class="citation book cs1">Urone, Paul Peter; Hinrichs, Roger (2022). <a rel="nofollow" class="external text" href="https://openstax.org/books/college-physics-2e/pages/29-7-probability-the-heisenberg-uncertainty-principle">"29.7 Probability: The Heisenberg Uncertainty Principle"</a>. College Physics 2e. <a href="/wiki/OpenStax" title="OpenStax">OpenStax</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=29.7+Probability%3A+The+Heisenberg+Uncertainty+Principle&rft.btitle=College+Physics+2e&rft.pub=OpenStax&rft.date=2022&rft.aulast=Urone&rft.aufirst=Paul+Peter&rft.au=Hinrichs%2C+Roger&rft_id=https%3A%2F%2Fopenstax.org%2Fbooks%2Fcollege-physics-2e%2Fpages%2F29-7-probability-the-heisenberg-uncertainty-principle&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-209"><a href="#cite_ref-209">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFItzyksonZuber1980" class="citation book cs1"><a href="/wiki/Claude_Itzykson" title="Claude Itzykson">Itzykson, C.</a>; <a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber, J.-B.</a> (1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4MwsAwAAQBAJ">Quantum Field Theory</a> (2005 ed.). Mineola, NY: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44568-7" title="Special:BookSources/978-0-486-44568-7"><bdi>978-0-486-44568-7</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2005053026">2005053026</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/61200849">61200849</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.place=Mineola%2C+NY&rft.edition=2005&rft.pub=Dover+Publications&rft.date=1980&rft_id=info%3Aoclcnum%2F61200849&rft_id=info%3Alccn%2F2005053026&rft.isbn=978-0-486-44568-7&rft.aulast=Itzykson&rft.aufirst=C.&rft.au=Zuber%2C+J.-B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4MwsAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-210"><a href="#cite_ref-210">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLow1971" class="citation book cs1">Low, Peter (1971). Classical Theory of Structures Based on the Differential Equation. Cambridge University Press. pp. 116–118. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-08089-7" title="Special:BookSources/978-0-521-08089-7"><bdi>978-0-521-08089-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Theory+of+Structures+Based+on+the+Differential+Equation&rft.pages=%3Cspan+class%3D%22nowrap%22%3E116-%3C%2Fspan%3E118&rft.pub=Cambridge+University+Press&rft.date=1971&rft.isbn=978-0-521-08089-7&rft.aulast=Low&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-211"><a href="#cite_ref-211">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBatchelor1967" class="citation book cs1">Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. p. 233. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66396-2" title="Special:BookSources/0-521-66396-2"><bdi>0-521-66396-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Fluid+Dynamics&rft.pages=233&rft.pub=Cambridge+University+Press&rft.date=1967&rft.isbn=0-521-66396-2&rft.aulast=Batchelor&rft.aufirst=G.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-A445-212">^ <a href="#cite_ref-A445_212-0">a</a> <a href="#cite_ref-A445_212-1">b</a> <a href="#cite_ref-A445_212-2">c</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 44–45 </li> <li id="cite_note-213"><a href="#cite_ref-213">^</a> <a rel="nofollow" class="external text" href="http://www.guinnessworldrecords.com/world-records/most-pi-places-memorised">"Most Pi Places Memorized"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160214205333/http://www.guinnessworldrecords.com/world-records/most-pi-places-memorised">Archived</a> 14 February 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Guinness World Records. </li> <li id="cite_note-japantimes-214"><a href="#cite_ref-japantimes_214-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOtake2006" class="citation news cs1">Otake, Tomoko (17 December 2006). <a rel="nofollow" class="external text" href="http://www.japantimes.co.jp/life/2006/12/17/general/how-can-anyone-remember-100000-numbers/">"How can anyone remember 100,000 numbers?"</a>. <a href="/wiki/The_Japan_Times" title="The Japan Times">The Japan Times</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130818004142/http://www.japantimes.co.jp/life/2006/12/17/life/how-can-anyone-remember-100000-numbers/">Archived</a> from the original on 18 August 2013. Retrieved 27 October 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Japan+Times&rft.atitle=How+can+anyone+remember+100%2C000+numbers%3F&rft.date=2006-12-17&rft.aulast=Otake&rft.aufirst=Tomoko&rft_id=http%3A%2F%2Fwww.japantimes.co.jp%2Flife%2F2006%2F12%2F17%2Fgeneral%2Fhow-can-anyone-remember-100000-numbers%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-215"><a href="#cite_ref-215">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDanesi2021" class="citation book cs1">Danesi, Marcel (January 2021). "Chapter 4: Pi in Popular Culture". Pi (π) in Nature, Art, and Culture. Brill. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tAsOEAAAQBAJ&pg=PA97">97</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1163%2F9789004433397">10.1163/9789004433397</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789004433373" title="Special:BookSources/9789004433373"><bdi>9789004433373</bdi></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:224869535">224869535</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4%3A+Pi+in+Popular+Culture&rft.btitle=Pi+%28%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E%29+in+Nature%2C+Art%2C+and+Culture&rft.pages=97&rft.pub=Brill&rft.date=2021-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A224869535%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1163%2F9789004433397&rft.isbn=9789004433373&rft.aulast=Danesi&rft.aufirst=Marcel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-216"><a href="#cite_ref-216">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRazPackard2009" class="citation journal cs1">Raz, A.; Packard, M.G. (2009). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4323087">"A slice of pi: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist"</a>. Neurocase. 15 (5): 361–372. <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%2F13554790902776896">10.1080/13554790902776896</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4323087">4323087</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19585350">19585350</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neurocase&rft.atitle=A+slice+of+pi%3A+An+exploratory+neuroimaging+study+of+digit+encoding+and+retrieval+in+a+superior+memorist&rft.volume=15&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E361-%3C%2Fspan%3E372&rft.date=2009&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4323087%23id-name%3DPMC&rft_id=info%3Apmid%2F19585350&rft_id=info%3Adoi%2F10.1080%2F13554790902776896&rft.aulast=Raz&rft.aufirst=A.&rft.au=Packard%2C+M.G.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4323087&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-217"><a href="#cite_ref-217">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeith" class="citation web cs1"><a href="/wiki/Mike_Keith_(mathematician)" title="Mike Keith (mathematician)">Keith, Mike</a>. <a rel="nofollow" class="external text" href="http://www.cadaeic.net/comments.htm">"Cadaeic Cadenza Notes & Commentary"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090118060210/http://cadaeic.net/comments.htm">Archived</a> from the original on 18 January 2009. Retrieved 29 July 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Cadaeic+Cadenza+Notes+%26+Commentary&rft.aulast=Keith&rft.aufirst=Mike&rft_id=http%3A%2F%2Fwww.cadaeic.net%2Fcomments.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-KeithNAW-218"><a href="#cite_ref-KeithNAW_218-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeithDiana_Keith2010" class="citation book cs1">Keith, Michael; Diana Keith (17 February 2010). Not A Wake: A dream embodying (pi)'s digits fully for 10,000 decimals. Vinculum Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9630097-1-5" title="Special:BookSources/978-0-9630097-1-5"><bdi>978-0-9630097-1-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Not+A+Wake%3A+A+dream+embodying+%28pi%29%27s+digits+fully+for+10%2C000+decimals&rft.pub=Vinculum+Press&rft.date=2010-02-17&rft.isbn=978-0-9630097-1-5&rft.aulast=Keith&rft.aufirst=Michael&rft.au=Diana+Keith&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-219"><a href="#cite_ref-219">^</a> For instance, Pickover calls π "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the <a href="/wiki/Parfums_Givenchy" title="Parfums Givenchy">Givenchy</a> π perfume, <a href="/wiki/Pi_(film)" title="Pi (film)">Pi (film)</a>, and <a href="/wiki/Pi_Day" title="Pi Day">Pi Day</a> as examples. See: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPickover1995" class="citation book cs1"><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Pickover, Clifford A.</a> (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/keystoinfinity00clif/page/59">Keys to Infinity</a>. Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/keystoinfinity00clif/page/59">59</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-11857-2" title="Special:BookSources/978-0-471-11857-2"><bdi>978-0-471-11857-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Keys+to+Infinity&rft.pages=59&rft.pub=Wiley+%26+Sons&rft.date=1995&rft.isbn=978-0-471-11857-2&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fkeystoinfinity00clif%2Fpage%2F59&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeterson2002" class="citation book cs1"><a href="/wiki/Ivars_Peterson" title="Ivars Peterson">Peterson, Ivars</a> (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4gWSAraVhtAC&pg=PA17">Mathematical Treks: From Surreal Numbers to Magic Circles</a>. MAA spectrum. Mathematical Association of America. p. 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-537-9" title="Special:BookSources/978-0-88385-537-9"><bdi>978-0-88385-537-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161129190818/https://books.google.com/books?id=4gWSAraVhtAC&pg=PA17">Archived</a> from the original on 29 November 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Treks%3A+From+Surreal+Numbers+to+Magic+Circles&rft.series=MAA+spectrum&rft.pages=17&rft.pub=Mathematical+Association+of+America&rft.date=2002&rft.isbn=978-0-88385-537-9&rft.aulast=Peterson&rft.aufirst=Ivars&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4gWSAraVhtAC%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-220"><a href="#cite_ref-220">^</a> <a href="#CITEREFPosamentierLehmann2004">Posamentier & Lehmann 2004</a>, p. 118 <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 50 </li> <li id="cite_note-221"><a href="#cite_ref-221">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, p. 14 </li> <li id="cite_note-222"><a href="#cite_ref-222">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPolsterRoss2012" class="citation book cs1"><a href="/wiki/Burkard_Polster" title="Burkard Polster">Polster, Burkard</a>; Ross, Marty (2012). Math Goes to the Movies. Johns Hopkins University Press. pp. 56–57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-421-40484-4" title="Special:BookSources/978-1-421-40484-4"><bdi>978-1-421-40484-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Math+Goes+to+the+Movies&rft.pages=%3Cspan+class%3D%22nowrap%22%3E56-%3C%2Fspan%3E57&rft.pub=Johns+Hopkins+University+Press&rft.date=2012&rft.isbn=978-1-421-40484-4&rft.aulast=Polster&rft.aufirst=Burkard&rft.au=Ross%2C+Marty&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-223"><a href="#cite_ref-223">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGill2005" class="citation journal cs1">Gill, Andy (4 November 2005). <a rel="nofollow" class="external text" href="http://gaffa.org/reaching/rev_aer_UK5.html">"Review of Aerial"</a>. <a href="/wiki/The_Independent" title="The Independent">The Independent</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061015122229/http://gaffa.org/reaching/rev_aer_UK5.html">Archived</a> from the original on 15 October 2006. <q>the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Independent&rft.atitle=Review+of+Aerial&rft.date=2005-11-04&rft.aulast=Gill&rft.aufirst=Andy&rft_id=http%3A%2F%2Fgaffa.org%2Freaching%2Frev_aer_UK5.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-224"><a href="#cite_ref-224">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRubillo1989" class="citation journal cs1">Rubillo, James M. (January 1989). "Disintegrate 'em". <a href="/wiki/The_Mathematics_Teacher" class="mw-redirect" title="The Mathematics Teacher">The Mathematics Teacher</a>. 82 (1): 10. <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/27966082">27966082</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=Disintegrate+%27em&rft.volume=82&rft.issue=1&rft.pages=10&rft.date=1989-01&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27966082%23id-name%3DJSTOR&rft.aulast=Rubillo&rft.aufirst=James+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-225"><a href="#cite_ref-225">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPetroski2011" class="citation book cs1"><a href="/wiki/Henry_Petroski" title="Henry Petroski">Petroski, Henry</a> (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oVXzxvS3MLUC&pg=PA47">Title An Engineer's Alphabet: Gleanings from the Softer Side of a Profession</a>. Cambridge University Press. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-50530-7" title="Special:BookSources/978-1-139-50530-7"><bdi>978-1-139-50530-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Title+An+Engineer%27s+Alphabet%3A+Gleanings+from+the+Softer+Side+of+a+Profession&rft.pages=47&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-1-139-50530-7&rft.aulast=Petroski&rft.aufirst=Henry&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoVXzxvS3MLUC%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-226"><a href="#cite_ref-226">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.usatoday.com/story/news/nation-now/2015/03/14/pi-day-kids-videos/24753169/">"Happy Pi Day! Watch these stunning videos of kids reciting 3.14"</a>. USAToday.com. 14 March 2015. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150315005038/http://www.usatoday.com/story/news/nation-now/2015/03/14/pi-day-kids-videos/24753169/">Archived</a> from the original on 15 March 2015. Retrieved 14 March 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=USAToday.com&rft.atitle=Happy+Pi+Day%21+Watch+these+stunning+videos+of+kids+reciting+3.14&rft.date=2015-03-14&rft_id=https%3A%2F%2Fwww.usatoday.com%2Fstory%2Fnews%2Fnation-now%2F2015%2F03%2F14%2Fpi-day-kids-videos%2F24753169%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-227"><a href="#cite_ref-227">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRosenthal2015" class="citation journal cs1">Rosenthal, Jeffrey S. (February 2015). <a rel="nofollow" class="external text" href="http://probability.ca/jeff/writing/PiInstant.html">"Pi Instant"</a>. Math Horizons. 22 (3): 22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Fmathhorizons.22.3.22">10.4169/mathhorizons.22.3.22</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:218542599">218542599</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math+Horizons&rft.atitle=Pi+Instant&rft.volume=22&rft.issue=3&rft.pages=22&rft.date=2015-02&rft_id=info%3Adoi%2F10.4169%2Fmathhorizons.22.3.22&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A218542599%23id-name%3DS2CID&rft.aulast=Rosenthal&rft.aufirst=Jeffrey+S.&rft_id=http%3A%2F%2Fprobability.ca%2Fjeff%2Fwriting%2FPiInstant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-228"><a href="#cite_ref-228">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriffin" class="citation news cs1">Griffin, Andrew. <a rel="nofollow" class="external text" href="https://www.independent.co.uk/news/science/pi-day-march-14-maths-google-doodle-pie-baking-celebrate-30-anniversary-a8254036.html">"Pi Day: Why some mathematicians refuse to celebrate 14 March and won't observe the dessert-filled day"</a>. The Independent. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190424151944/https://www.independent.co.uk/news/science/pi-day-march-14-maths-google-doodle-pie-baking-celebrate-30-anniversary-a8254036.html">Archived</a> from the original on 24 April 2019. Retrieved 2 February 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Independent&rft.atitle=Pi+Day%3A+Why+some+mathematicians+refuse+to+celebrate+14+March+and+won%27t+observe+the+dessert-filled+day&rft.aulast=Griffin&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fwww.independent.co.uk%2Fnews%2Fscience%2Fpi-day-march-14-maths-google-doodle-pie-baking-celebrate-30-anniversary-a8254036.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-229"><a href="#cite_ref-229">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFreibergerThomas2015" class="citation book cs1">Freiberger, Marianne; Thomas, Rachel (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IbR-BAAAQBAJ&pg=PT133">"Tau – the new π"</a>. Numericon: A Journey through the Hidden Lives of Numbers. Quercus. p. 159. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-62365-411-5" title="Special:BookSources/978-1-62365-411-5"><bdi>978-1-62365-411-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tau+%E2%80%93+the+new+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.btitle=Numericon%3A+A+Journey+through+the+Hidden+Lives+of+Numbers&rft.pages=159&rft.pub=Quercus&rft.date=2015&rft.isbn=978-1-62365-411-5&rft.aulast=Freiberger&rft.aufirst=Marianne&rft.au=Thomas%2C+Rachel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIbR-BAAAQBAJ%26pg%3DPT133&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-230"><a href="#cite_ref-230">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbbott2012" class="citation journal cs1">Abbott, Stephen (April 2012). <a rel="nofollow" class="external text" href="http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf">"My Conversion to Tauism"</a> (PDF). Math Horizons. 19 (4): 34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Fmathhorizons.19.4.34">10.4169/mathhorizons.19.4.34</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:126179022">126179022</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130928095819/http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf">Archived</a> (PDF) from the original on 28 September 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math+Horizons&rft.atitle=My+Conversion+to+Tauism&rft.volume=19&rft.issue=4&rft.pages=34&rft.date=2012-04&rft_id=info%3Adoi%2F10.4169%2Fmathhorizons.19.4.34&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126179022%23id-name%3DS2CID&rft.aulast=Abbott&rft.aufirst=Stephen&rft_id=http%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2FMathhorizons%2Fapr12_aftermath.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-231"><a href="#cite_ref-231">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPalais2001" class="citation journal cs1">Palais, Robert (2001). <a rel="nofollow" class="external text" href="http://www.math.utah.edu/~palais/pi.pdf">"π Is Wrong!"</a> (PDF). The Mathematical Intelligencer. 23 (3): 7–8. <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%2FBF03026846">10.1007/BF03026846</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:120965049">120965049</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120622070009/http://www.math.utah.edu/~palais/pi.pdf">Archived</a> (PDF) from the original on 22 June 2012.</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=%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E+Is+Wrong%21&rft.volume=23&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E7-%3C%2Fspan%3E8&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2FBF03026846&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120965049%23id-name%3DS2CID&rft.aulast=Palais&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.math.utah.edu%2F~palais%2Fpi.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-232"><a href="#cite_ref-232">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130713084345/http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp">"Life of pi in no danger – Experts cold-shoulder campaign to replace with tau"</a>. Telegraph India. 30 June 2011. Archived from <a rel="nofollow" class="external text" href="http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp">the original</a> on 13 July 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Telegraph+India&rft.atitle=Life+of+pi+in+no+danger+%E2%80%93+Experts+cold-shoulder+campaign+to+replace+with+tau&rft.date=2011-06-30&rft_id=http%3A%2F%2Fwww.telegraphindia.com%2F1110630%2Fjsp%2Fnation%2Fstory_14178997.jsp&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-233"><a href="#cite_ref-233">^</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://www.sciencenews.org/blog/science-the-public/forget-pi-day-we-should-be-celebrating-tau-day">"Forget Pi Day. We should be celebrating Tau Day | Science News"</a>. Retrieved 2 May 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Forget+Pi+Day.+We+should+be+celebrating+Tau+Day+%7C+Science+News&rft_id=https%3A%2F%2Fwww.sciencenews.org%2Fblog%2Fscience-the-public%2Fforget-pi-day-we-should-be-celebrating-tau-day&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-234"><a href="#cite_ref-234">^</a> <a href="#CITEREFArndtHaenel2006">Arndt & Haenel 2006</a>, pp. 211–212 <a href="#CITEREFPosamentierLehmann2004">Posamentier & Lehmann 2004</a>, pp. 36–37 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHallerberg1977" class="citation journal cs1">Hallerberg, Arthur (May 1977). "Indiana's squared circle". Mathematics Magazine. 50 (3): 136–140. <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%2F2689499">10.2307/2689499</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/2689499">2689499</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Indiana%27s+squared+circle&rft.volume=50&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E136-%3C%2Fspan%3E140&rft.date=1977-05&rft_id=info%3Adoi%2F10.2307%2F2689499&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2689499%23id-name%3DJSTOR&rft.aulast=Hallerberg&rft.aufirst=Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> </li> <li id="cite_note-235"><a href="#cite_ref-235">^</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> (3 October 1990). <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. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160413230304/http://www.ntg.nl/maps/05/34.pdf">Archived</a> (PDF) from the original on 13 April 2016. Retrieved 17 February 2017.</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%3APi" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="General_and_cited_sources">General and cited sources</h3></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbramson2014" class="citation book cs1">Abramson, Jay (2014). <a rel="nofollow" class="external text" href="https://openstax.org/details/books/precalculus">Precalculus</a>. <a href="/wiki/OpenStax" title="OpenStax">OpenStax</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus&rft.pub=OpenStax&rft.date=2014&rft.aulast=Abramson&rft.aufirst=Jay&rft_id=https%3A%2F%2Fopenstax.org%2Fdetails%2Fbooks%2Fprecalculus&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAndrewsAskeyRoy1999" class="citation book cs1">Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kGshpCa3eYwC&pg=PA59">Special Functions</a>. Cambridge: University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78988-2" title="Special:BookSources/978-0-521-78988-2"><bdi>978-0-521-78988-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+Functions&rft.place=Cambridge&rft.pub=University+Press&rft.date=1999&rft.isbn=978-0-521-78988-2&rft.aulast=Andrews&rft.aufirst=George+E.&rft.au=Askey%2C+Richard&rft.au=Roy%2C+Ranjan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkGshpCa3eYwC%26pg%3DPA59&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArndtHaenel2006" class="citation book cs1">Arndt, Jörg; Haenel, Christoph (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QwwcmweJCDQC">Pi Unleashed</a>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66572-4" title="Special:BookSources/978-3-540-66572-4"><bdi>978-3-540-66572-4</bdi></a>. Retrieved 5 June 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+Unleashed&rft.pub=Springer-Verlag&rft.date=2006&rft.isbn=978-3-540-66572-4&rft.aulast=Arndt&rft.aufirst=J%C3%B6rg&rft.au=Haenel%2C+Christoph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQwwcmweJCDQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> English translation by Catriona and David Lischka.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBerggrenBorweinBorwein1997" class="citation book cs1">Berggren, Lennart; <a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan</a>; <a href="/wiki/Peter_Borwein" title="Peter Borwein">Borwein, Peter</a> (1997). Pi: a Source Book. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20571-7" title="Special:BookSources/978-0-387-20571-7"><bdi>978-0-387-20571-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi%3A+a+Source+Book&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=978-0-387-20571-7&rft.aulast=Berggren&rft.aufirst=Lennart&rft.au=Borwein%2C+Jonathan&rft.au=Borwein%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoyerMerzbach1991" class="citation book cs1">Boyer, Carl B.; <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2 ed.). Wiley. <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.edition=2&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+B.&rft.au=Merzbach%2C+Uta+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBronshteĭnSemendiaev1971" class="citation book cs1">Bronshteĭn, Ilia; Semendiaev, K.A. (1971). <a href="/wiki/A_Guide_Book_to_Mathematics" class="mw-redirect" title="A Guide Book to Mathematics">A Guide Book to Mathematics</a>. <a href="/wiki/Verlag_Harri_Deutsch" title="Verlag Harri Deutsch">Verlag Harri Deutsch</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-87144-095-3" title="Special:BookSources/978-3-87144-095-3"><bdi>978-3-87144-095-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+Book+to+Mathematics&rft.pub=Verlag+Harri+Deutsch&rft.date=1971&rft.isbn=978-3-87144-095-3&rft.aulast=Bronshte%C4%ADn&rft.aufirst=Ilia&rft.au=Semendiaev%2C+K.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDymMcKean1972" class="citation book cs1">Dym, H.; McKean, H. P. (1972). Fourier series and integrals. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+series+and+integrals&rft.pub=Academic+Press&rft.date=1972&rft.aulast=Dym&rft.aufirst=H.&rft.au=McKean%2C+H.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><cite class="citation wikicite" id="CITEREFEymardLafon2004"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">Eymard, Pierre; Lafon, Jean Pierre (2004). The Number π. Translated by Wilson, Stephen. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3246-2" title="Special:BookSources/978-0-8218-3246-2"><bdi>978-0-8218-3246-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Number+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.pub=American+Mathematical+Society&rft.date=2004&rft.isbn=978-0-8218-3246-2&rft.aulast=Eymard&rft.aufirst=Pierre&rft.au=Lafon%2C+Jean+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"> English translation of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1 cs1-prop-foreign-lang-source">Autour du nombre π (in French). Hermann. 1999.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Autour+du+nombre+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.pub=Hermann&rft.date=1999&rft.aulast=Eymard&rft.aufirst=Pierre&rft.au=Lafon%2C+Jean+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></cite></li> <li><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://archive.org/details/pi00alfr_0">π: A Biography of the World's Most Mysterious Number</a>. Prometheus Books. <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=%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E%3A+A+Biography+of+the+World%27s+Most+Mysterious+Number&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%2Farchive.org%2Fdetails%2Fpi00alfr_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRemmert2012" class="citation book cs1">Remmert, Reinhold (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z53SBwAAQBAJ&pg=PA123">"Ch. 5 What is π?"</a>. In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.). Numbers. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-1005-4" title="Special:BookSources/978-1-4612-1005-4"><bdi>978-1-4612-1005-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ch.+5+What+is+%CF%80%3F&rft.btitle=Numbers&rft.pub=Springer&rft.date=2012&rft.isbn=978-1-4612-1005-4&rft.aulast=Remmert&rft.aufirst=Reinhold&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZ53SBwAAQBAJ%26pg%3DPA123&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316" /><div class="refbegin refbegin-hanging-indents" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBlatner1999" class="citation book cs1">Blatner, David (1999). The Joy of π. Walker & Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8027-7562-7" title="Special:BookSources/978-0-8027-7562-7"><bdi>978-0-8027-7562-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Joy+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.pub=Walker+%26+Company&rft.date=1999&rft.isbn=978-0-8027-7562-7&rft.aulast=Blatner&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDelahaye1997" class="citation book cs1"><a href="/wiki/Jean-Paul_Delahaye" title="Jean-Paul Delahaye">Delahaye, Jean-Paul</a> (1997). Le fascinant nombre π. Paris: Bibliothèque Pour la Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/2-902918-25-9" title="Special:BookSources/2-902918-25-9"><bdi>2-902918-25-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le+fascinant+nombre+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E&rft.place=Paris&rft.pub=Biblioth%C3%A8que+Pour+la+Science&rft.date=1997&rft.isbn=2-902918-25-9&rft.aulast=Delahaye&rft.aufirst=Jean-Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Pi" class="extiw" title="commons:Category:Pi">Pi</a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Pi.html">"Pi"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Pi&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APi" class="Z3988"></li> <li>Demonstration by Lambert (1761) of irrationality of π, <a rel="nofollow" class="external text" href="https://www.bibnum.education.fr/mathematiques/theorie-des-nombres/lambert-et-l-irrationalite-de-p-1761">online</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141231045534/https://www.bibnum.education.fr/mathematiques/theorie-des-nombres/lambert-et-l-irrationalite-de-p-1761">Archived</a> 31 December 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> and analysed <a rel="nofollow" class="external text" href="https://www.bibnum.education.fr/sites/default/files/24-lambert-analysis.pdf">BibNum</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150402115151/https://www.bibnum.education.fr/sites/default/files/24-lambert-analysis.pdf">Archived</a> 2 April 2015 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (PDF).</li> <li><a rel="nofollow" class="external text" href="https://pisearch.org/pi">π Search Engine</a> 2 billion searchable digits of π, e and √2</li> <li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/kwty4hsz">approximation von π by lattice points</a> and <a rel="nofollow" class="external text" href="https://www.geogebra.org/m/bxfa364u">approximation of π with rectangles and trapezoids</a> (interactive illustrations)</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" 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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> (σ)</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 href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's</a> (e)</li> <li><a class="mw-selflink selflink">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" 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class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.eqiad.main-58c4b96c94-6z89r","wgBackendResponseTime":477,"wgPageParseReport":{"limitreport":{"cputime":"3.933","walltime":"4.422","ppvisitednodes":{"value":27186,"limit":1000000},"postexpandincludesize":{"value":448188,"limit":2097152},"templateargumentsize":{"value":25438,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":22,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":688637,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 3580.734 1 -total"," 44.34% 1587.747 2 Template:Reflist"," 20.26% 725.293 90 Template:Cite_book"," 13.62% 487.521 48 Template:Cite_journal"," 9.86% 353.182 70 Template:Sfn"," 8.39% 300.477 185 Template:Math"," 4.60% 164.750 1 Template:Short_description"," 4.37% 156.621 2 Template:Lang"," 3.39% 121.259 2 Template:Pagetype"," 3.26% 116.682 15 Template:Cite_web"]},"scribunto":{"limitreport-timeusage":{"value":"2.385","limit":"10.000"},"limitreport-memusage":{"value":26305868,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAaboe1964\"] = 1,\n [\"CITEREFAbbott2012\"] = 1,\n [\"CITEREFAbramson2014\"] = 1,\n [\"CITEREFAhlfors1966\"] = 2,\n [\"CITEREFAndrewsAskeyRoy1999\"] = 1,\n [\"CITEREFApostol1967\"] = 1,\n [\"CITEREFArchibald1921\"] = 1,\n [\"CITEREFArndtHaenel2006\"] = 1,\n [\"CITEREFArtin1964\"] = 1,\n [\"CITEREFAzarian2010\"] = 1,\n [\"CITEREFBailey2003\"] = 1,\n [\"CITEREFBaileyBorwein2016\"] = 1,\n [\"CITEREFBaileyBorweinPlouffe1997\"] = 1,\n [\"CITEREFBaileyPlouffeBorweinBorwein1997\"] = 1,\n [\"CITEREFBaltzer1870\"] = 1,\n [\"CITEREFBarrow1860\"] = 1,\n [\"CITEREFBatchelor1967\"] = 1,\n [\"CITEREFBeckmann1989\"] = 1,\n [\"CITEREFBellard\"] = 1,\n [\"CITEREFBenjamin_NillAndreas_Paffenholz2014\"] = 1,\n [\"CITEREFBerggrenBorweinBorwein1997\"] = 1,\n [\"CITEREFBlatner1999\"] = 1,\n [\"CITEREFBorwein2014\"] = 1,\n [\"CITEREFBorweinBorwein1987\"] = 1,\n [\"CITEREFBorweinBorwein1988\"] = 1,\n [\"CITEREFBorweinBorweinDilcher1989\"] = 1,\n [\"CITEREFBourbaki1979\"] = 1,\n [\"CITEREFBourbaki1981\"] = 1,\n [\"CITEREFBoyerMerzbach1991\"] = 1,\n [\"CITEREFBrezinski2009\"] = 1,\n [\"CITEREFBronshteĭnSemendiaev1971\"] = 1,\n [\"CITEREFCajori1913\"] = 1,\n [\"CITEREFCajori2007\"] = 1,\n [\"CITEREFCassel2022\"] = 1,\n [\"CITEREFChavel2001\"] = 1,\n [\"CITEREFChien-Lih2004\"] = 1,\n [\"CITEREFChien-Lih2005\"] = 1,\n [\"CITEREFConnor2010\"] = 1,\n [\"CITEREFCooker2011\"] = 1,\n [\"CITEREFDanesi2021\"] = 1,\n [\"CITEREFDel_PinoDolbeault2002\"] = 1,\n [\"CITEREFDelahaye1997\"] = 1,\n [\"CITEREFDymMcKean1972\"] = 1,\n [\"CITEREFEhlers2000\"] = 1,\n [\"CITEREFEuler1727\"] = 1,\n [\"CITEREFEuler1736\"] = 1,\n [\"CITEREFEuler1747\"] = 1,\n [\"CITEREFEuler1755\"] = 1,\n [\"CITEREFEuler1798\"] = 1,\n [\"CITEREFEuler1922\"] = 1,\n [\"CITEREFEvans1997\"] = 1,\n [\"CITEREFEymardLafon2004\"] = 1,\n [\"CITEREFFolland1989\"] = 1,\n [\"CITEREFFreibergerThomas2015\"] = 1,\n [\"CITEREFGibbons2006\"] = 1,\n [\"CITEREFGilbargTrudinger1983\"] = 1,\n [\"CITEREFGill2005\"] = 1,\n [\"CITEREFGregorius1695\"] = 1,\n [\"CITEREFGrienbergerus1630\"] = 1,\n [\"CITEREFGriffin\"] = 1,\n [\"CITEREFGrünbaum1960\"] = 1,\n [\"CITEREFGupta1992\"] = 1,\n [\"CITEREFHallerberg1977\"] = 1,\n [\"CITEREFHallidayResnickWalker1997\"] = 1,\n [\"CITEREFHardy2008\"] = 1,\n [\"CITEREFHaruka_Iwao2019\"] = 1,\n [\"CITEREFHayes2014\"] = 1,\n [\"CITEREFHermanStrang2016\"] = 1,\n [\"CITEREFHerz-Fischler2000\"] = 1,\n [\"CITEREFHilbertCourant1966\"] = 1,\n [\"CITEREFHorvath1983\"] = 1,\n [\"CITEREFHowe1980\"] = 1,\n [\"CITEREFItzyksonZuber1980\"] = 1,\n [\"CITEREFJoglekar2005\"] = 1,\n [\"CITEREFJones1706\"] = 1,\n [\"CITEREFJoseph1991\"] = 1,\n [\"CITEREFKeith\"] = 1,\n [\"CITEREFKeithDiana_Keith2010\"] = 1,\n [\"CITEREFKennedy1978\"] = 1,\n [\"CITEREFKlebanoff2001\"] = 1,\n [\"CITEREFKnuth1990\"] = 1,\n [\"CITEREFKobayashiNomizu1996\"] = 1,\n [\"CITEREFKontsevichZagier2001\"] = 1,\n [\"CITEREFL._EspositoC._NitschC._Trombetti2011\"] = 1,\n [\"CITEREFLandau1934\"] = 1,\n [\"CITEREFLange1999\"] = 1,\n [\"CITEREFLehmer1938\"] = 1,\n [\"CITEREFLindemann1882\"] = 1,\n [\"CITEREFLow1971\"] = 1,\n [\"CITEREFMaor2009\"] = 1,\n [\"CITEREFMartiniMontejanoOliveros2019\"] = 1,\n [\"CITEREFMollin1999\"] = 1,\n [\"CITEREFMumford1983\"] = 1,\n [\"CITEREFMurtyRath2014\"] = 1,\n [\"CITEREFNewton1971\"] = 1,\n [\"CITEREFNicholsonJeenel1955\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson1999\"] = 1,\n [\"CITEREFOgilvyAnderson1988\"] = 1,\n [\"CITEREFOtake2006\"] = 1,\n [\"CITEREFOughtred1648\"] = 1,\n [\"CITEREFOughtred1652\"] = 1,\n [\"CITEREFOughtred1694\"] = 1,\n [\"CITEREFOvsienkoTabachnikov2004\"] = 1,\n [\"CITEREFPalais2001\"] = 1,\n [\"CITEREFPalmer2010\"] = 1,\n [\"CITEREFPayneWeinberger1960\"] = 1,\n [\"CITEREFPeitgen2004\"] = 1,\n [\"CITEREFPeterson2002\"] = 1,\n [\"CITEREFPetroski2011\"] = 1,\n [\"CITEREFPickover1995\"] = 1,\n [\"CITEREFPlatonovRapinchuk1994\"] = 1,\n [\"CITEREFPlofker2009\"] = 1,\n [\"CITEREFPlouffe2006\"] = 1,\n [\"CITEREFPlouffe2022\"] = 1,\n [\"CITEREFPolsterRoss2012\"] = 1,\n [\"CITEREFPortStone1978\"] = 1,\n [\"CITEREFPosamentierLehmann2004\"] = 1,\n [\"CITEREFRabinowitzWagon1995\"] = 1,\n [\"CITEREFRamaley1969\"] = 1,\n [\"CITEREFRazPackard2009\"] = 1,\n [\"CITEREFReitwiesner1950\"] = 1,\n [\"CITEREFRemmert2012\"] = 1,\n [\"CITEREFRosenthal2015\"] = 1,\n [\"CITEREFRoy1990\"] = 1,\n [\"CITEREFRoy2021\"] = 1,\n [\"CITEREFRubillo1989\"] = 1,\n [\"CITEREFRudin1976\"] = 1,\n [\"CITEREFRudin1986\"] = 1,\n [\"CITEREFSalikhov2008\"] = 1,\n [\"CITEREFSandifer2009\"] = 1,\n [\"CITEREFSandifer2014\"] = 1,\n [\"CITEREFSchey1996\"] = 1,\n [\"CITEREFSchlagerLauer2001\"] = 1,\n [\"CITEREFSchudel2009\"] = 1,\n [\"CITEREFSegner1756\"] = 1,\n [\"CITEREFSegner1761\"] = 1,\n [\"CITEREFSmith1929\"] = 1,\n [\"CITEREFSmith1958\"] = 1,\n [\"CITEREFSondow1994\"] = 1,\n [\"CITEREFSpivak1999\"] = 1,\n [\"CITEREFStein1970\"] = 1,\n [\"CITEREFSteinWeiss1971\"] = 1,\n [\"CITEREFT._FriedmannC.R._Hagen2015\"] = 1,\n [\"CITEREFTalenti1976\"] = 1,\n [\"CITEREFTate1950\"] = 1,\n [\"CITEREFThompson1894\"] = 1,\n [\"CITEREFTitchmarsh1948\"] = 1,\n [\"CITEREFTweddle1991\"] = 1,\n [\"CITEREFUroneHinrichs2022\"] = 1,\n [\"CITEREFVieta1593\"] = 1,\n [\"CITEREFWaldschmidt2021\"] = 1,\n [\"CITEREFWeierstrass1841\"] = 1,\n [\"CITEREFWeisstein\"] = 4,\n [\"CITEREFWells1997\"] = 1,\n [\"CITEREFYeo2006\"] = 1,\n [\"CITEREFYoder1996\"] = 1,\n [\"tau\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 4,\n [\"'\"] = 1,\n [\"3\"] = 1,\n [\"4\"] = 3,\n [\"=\"] = 5,\n [\"About\"] = 1,\n [\"Abs\"] = 1,\n [\"Anchor\"] = 1,\n [\"Authority control\"] = 1,\n [\"Br\"] = 10,\n [\"Citation\"] = 2,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 90,\n [\"Cite conference\"] = 1,\n [\"Cite journal\"] = 48,\n [\"Cite magazine\"] = 2,\n [\"Cite news\"] = 6,\n [\"Cite web\"] = 15,\n [\"Closed-closed\"] = 2,\n [\"Commons category\"] = 1,\n [\"Comparison_pi_infinite_series.svg\"] = 1,\n [\"Efn\"] = 3,\n [\"Excerpt\"] = 1,\n [\"Featured article\"] = 1,\n [\"Gaps\"] = 4,\n [\"Harvid\"] = 1,\n [\"Harvnb\"] = 43,\n [\"Harvtxt\"] = 1,\n [\"IPAc-en\"] = 2,\n [\"Irrational number\"] = 1,\n [\"Lang\"] = 2,\n [\"Main\"] = 3,\n [\"Math\"] = 181,\n [\"Mathworld\"] = 1,\n [\"Mset\"] = 1,\n [\"Multiple image\"] = 2,\n [\"Mvar\"] = 27,\n [\"Nbsp\"] = 1,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 2,\n [\"OEIS2C\"] = 8,\n [\"Pb\"] = 2,\n [\"Pi\"] = 273,\n [\"Pi box\"] = 1,\n [\"Pp\"] = 1,\n [\"Quote box\"] = 1,\n [\"Radic\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"Respell\"] = 1,\n [\"See also\"] = 3,\n [\"Sfn\"] = 70,\n [\"Sfrac\"] = 19,\n [\"Short description\"] = 1,\n [\"Sqrt\"] = 1,\n [\"Sup\"] = 8,\n [\"TOC limit\"] = 1,\n [\"Tmath\"] = 1,\n [\"Use Oxford spelling\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 5,\n [\"Wikicite\"] = 1,\n [\"′\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n","limitreport-profile":[["?","360","15.3"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","280","11.9"],["dataWrapper \u003Cmw.lua:672\u003E","180","7.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","160","6.8"],["recursiveClone \u003CmwInit.lua:45\u003E","160","6.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","140","5.9"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","120","5.1"],["\u003Cmw.lua:694\u003E","80","3.4"],["(for 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