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<li id="toc-Ancient_Greece" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient_Greece"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Ancient Greece</span> </div> </a> <ul id="toc-Ancient_Greece-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-India" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#India"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>India</span> </div> </a> <ul id="toc-India-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Islamic_World" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Islamic_World"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Islamic World</span> </div> </a> <ul id="toc-Islamic_World-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_period"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Modern period</span> </div> </a> <ul id="toc-Modern_period-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Square_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Square roots</span> </div> </a> <ul id="toc-Square_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>General roots</span> </div> </a> <ul id="toc-General_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Logarithms</span> </div> </a> <ul id="toc-Logarithms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Algebraic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Algebraic</span> </div> </a> <ul id="toc-Algebraic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transcendental" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transcendental"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Transcendental</span> </div> </a> <ul id="toc-Transcendental-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decimal_expansions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Decimal_expansions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Decimal expansions</span> </div> </a> <ul id="toc-Decimal_expansions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irrational_powers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Irrational_powers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Irrational powers</span> </div> </a> <ul id="toc-Irrational_powers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Open_questions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Open_questions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Open questions</span> </div> </a> <ul id="toc-Open_questions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_constructive_mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_constructive_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In constructive mathematics</span> </div> </a> <ul id="toc-In_constructive_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_of_all_irrationals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Set_of_all_irrationals"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Set of all irrationals</span> </div> </a> <ul id="toc-Set_of_all_irrationals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Irrational number</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 89 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-89" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">89 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Irrasionale_getal" title="Irrasionale getal – Afrikaans" lang="af" hreflang="af" data-title="Irrasionale getal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Irrationaalloho" title="Irrationaalloho – Inari Sami" lang="smn" hreflang="smn" data-title="Irrationaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%BA%D9%8A%D8%B1_%D9%83%D8%B3%D8%B1%D9%8A" title="عدد غير كسري – Arabic" lang="ar" hreflang="ar" data-title="عدد غير كسري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A7%B0%E0%A6%BF%E0%A6%AE%E0%A7%87%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="অপৰিমেয় সংখ্যা – Assamese" lang="as" hreflang="as" data-title="অপৰিমেয় সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_irracional" title="Númberu irracional – Asturian" lang="ast" hreflang="ast" data-title="Númberu irracional" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0rrasional_%C9%99d%C9%99dl%C9%99r" title="İrrasional ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="İrrasional ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AE%E0%A7%82%E0%A6%B2%E0%A6%A6_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="অমূলদ সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="অমূলদ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%BB-l%C3%AD-s%C3%B2%CD%98" title="Bû-lí-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Bû-lí-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C_%D2%BB%D0%B0%D0%BD" title="Иррациональ һан – Bashkir" lang="ba" hreflang="ba" data-title="Иррациональ һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D1%80%D0%B0%D1%86%D1%8B%D1%8F%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Ірацыянальны лік – Belarusian" lang="be" hreflang="be" data-title="Ірацыянальны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Ирационално число – Bulgarian" lang="bg" hreflang="bg" data-title="Ирационално число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Iracionalan_broj" title="Iracionalan broj – Bosnian" lang="bs" hreflang="bs" data-title="Iracionalan broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_irracional" title="Nombre irracional – Catalan" lang="ca" hreflang="ca" data-title="Nombre irracional" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Иррационаллă хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Иррационаллă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Iracion%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Iracionální číslo – Czech" lang="cs" hreflang="cs" data-title="Iracionální číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_anghymarebol" title="Rhif anghymarebol – Welsh" lang="cy" hreflang="cy" data-title="Rhif anghymarebol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Irrationale_tal" title="Irrationale tal – Danish" lang="da" hreflang="da" data-title="Irrationale tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Irrationale_Zahl" title="Irrationale Zahl – German" lang="de" hreflang="de" data-title="Irrationale Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Irratsionaalarvud" title="Irratsionaalarvud – Estonian" lang="et" hreflang="et" data-title="Irratsionaalarvud" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CF%81%CF%81%CE%B7%CF%84%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Άρρητος αριθμός – Greek" lang="el" hreflang="el" data-title="Άρρητος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – Spanish" lang="es" hreflang="es" data-title="Número irracional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Neracionala_nombro" title="Neracionala nombro – Esperanto" lang="eo" hreflang="eo" data-title="Neracionala nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_irrazional" title="Zenbaki irrazional – Basque" lang="eu" hreflang="eu" data-title="Zenbaki irrazional" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%DA%AF%D9%86%DA%AF" title="عدد گنگ – Persian" lang="fa" hreflang="fa" data-title="عدد گنگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Irrationell_t%C3%B8l" title="Irrationell tøl – Faroese" lang="fo" hreflang="fo" data-title="Irrationell tøl" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fr.wikipedia.org/wiki/Nombre_irrationnel" title="Nombre irrationnel – French" lang="fr" hreflang="fr" data-title="Nombre irrationnel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_%C3%A9ag%C3%B3imheasta" title="Uimhir éagóimheasta – Irish" lang="ga" hreflang="ga" data-title="Uimhir éagóimheasta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – Galician" lang="gl" hreflang="gl" data-title="Número irracional" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수 – Korean" lang="ko" hreflang="ko" data-title="무리수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%BC%D5%A1%D6%81%D5%AB%D5%B8%D5%B6%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Իռացիոնալ թիվ – Armenian" lang="hy" hreflang="hy" data-title="Իռացիոնալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अपरिमेय संख्या – Hindi" lang="hi" hreflang="hi" data-title="अपरिमेय संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Iracionalni_broj" title="Iracionalni broj – Croatian" lang="hr" hreflang="hr" data-title="Iracionalni broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Neracionala_nombro" title="Neracionala nombro – Ido" lang="io" hreflang="io" data-title="Neracionala nombro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_irasional" title="Bilangan irasional – Indonesian" lang="id" hreflang="id" data-title="Bilangan irasional" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%93r%C3%A6%C3%B0ar_t%C3%B6lur" title="Óræðar tölur – Icelandic" lang="is" hreflang="is" data-title="Óræðar tölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_irrazionale" title="Numero irrazionale – Italian" lang="it" hreflang="it" data-title="Numero irrazionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%90%D7%99-%D7%A8%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99" title="מספר אי-רציונלי – Hebrew" lang="he" hreflang="he" data-title="מספר אי-רציונלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%A0%E1%83%90%E1%83%AA%E1%83%98%E1%83%9D%E1%83%9C%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ირაციონალური რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="ირაციონალური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B1%D0%B0%D0%B9%D1%81%D1%8B%D0%B7_%D1%81%D0%B0%D0%BD" title="Рабайсыз сан – Kazakh" lang="kk" hreflang="kk" data-title="Рабайсыз сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_isiyowiana" title="Namba isiyowiana – Swahili" lang="sw" hreflang="sw" data-title="Namba isiyowiana" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_na_aql%C3%AE" title="Hejmarên na aqlî – Kurdish" lang="ku" hreflang="ku" data-title="Hejmarên na aqlî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B4%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Иррационалдык сан – Kyrgyz" lang="ky" hreflang="ky" data-title="Иррационалдык сан" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AD%E0%BA%B0%E0%BA%9B%E0%BA%BB%E0%BA%81%E0%BA%81%E0%BA%B0%E0%BA%95%E0%BA%B4" title="ຈຳນວນອະປົກກະຕິ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນອະປົກກະຕິ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_irrationalis" title="Numerus irrationalis – Latin" lang="la" hreflang="la" data-title="Numerus irrationalis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Iracion%C4%81ls_skaitlis" title="Iracionāls skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Iracionāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Iracionalusis_skai%C4%8Dius" title="Iracionalusis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Iracionalusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_irazziunaal" title="Nümar irazziunaal – Lombard" lang="lmo" hreflang="lmo" data-title="Nümar irazziunaal" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Irracion%C3%A1lis_sz%C3%A1mok" title="Irracionális számok – Hungarian" lang="hu" hreflang="hu" data-title="Irracionális számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Ирационален број – Macedonian" lang="mk" hreflang="mk" data-title="Ирационален број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsivoasaina" title="Isa tsivoasaina – Malagasy" lang="mg" hreflang="mg" data-title="Isa tsivoasaina" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%AD%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%A8%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="അഭിന്നകസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="അഭിന്നകസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Numru_irrazzjonali" title="Numru irrazzjonali – Maltese" lang="mt" hreflang="mt" data-title="Numru irrazzjonali" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अपरिमेय संख्या – Marathi" lang="mr" hreflang="mr" data-title="अपरिमेय संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_bukan_nisbah" title="Nombor bukan nisbah – Malay" lang="ms" hreflang="ms" data-title="Nombor bukan nisbah" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE" title="Иррационал тоо – Mongolian" lang="mn" hreflang="mn" data-title="Иррационал тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Irrationaal_getal" title="Irrationaal getal – Dutch" lang="nl" hreflang="nl" data-title="Irrationaal getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%84%A1%E7%90%86%E6%95%B0" title="無理数 – Japanese" lang="ja" hreflang="ja" data-title="無理数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Irrasjonalt_tall" title="Irrasjonalt tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Irrasjonalt tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Irrasjonale_tal" title="Irrasjonale tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Irrasjonale tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%85%E0%AC%AA%E0%AC%B0%E0%AC%BF%E0%AC%AE%E0%AD%87%E0%AD%9F_%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE" title="ଅପରିମେୟ ସଂଖ୍ୟା – Odia" lang="or" hreflang="or" data-title="ଅପରିମେୟ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Irratsional_sonlar" title="Irratsional sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Irratsional sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%BC%E0%A9%88%E0%A8%B0-%E0%A8%AC%E0%A8%9F%E0%A9%87%E0%A8%A8%E0%A9%81%E0%A8%AE%E0%A8%BE_%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86" title="ਗ਼ੈਰ-ਬਟੇਨੁਮਾ ਸੰਖਿਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਗ਼ੈਰ-ਬਟੇਨੁਮਾ ਸੰਖਿਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_niewymierne" title="Liczby niewymierne – Polish" lang="pl" hreflang="pl" data-title="Liczby niewymierne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – Portuguese" lang="pt" hreflang="pt" data-title="Número irracional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_ira%C8%9Bional" title="Număr irațional – Romanian" lang="ro" hreflang="ro" data-title="Număr irațional" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Иррациональное число – Russian" lang="ru" hreflang="ru" data-title="Иррациональное число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_irracional%C3%AB" title="Numrat irracionalë – Albanian" lang="sq" hreflang="sq" data-title="Numrat irracionalë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_irrazziunali" title="Nùmmuru irrazziunali – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru irrazziunali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Irrational_number" title="Irrational number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Irrational number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Iracion%C3%A1lne_%C4%8D%C3%ADslo" title="Iracionálne číslo – Slovak" lang="sk" hreflang="sk" data-title="Iracionálne číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Iracionalno_%C5%A1tevilo" title="Iracionalno število – Slovenian" lang="sl" hreflang="sl" data-title="Iracionalno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D9%86%D8%A7%DA%95%DB%8E%DA%98%DB%95%DB%8C%DB%8C" title="ژمارەی ناڕێژەیی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی ناڕێژەیی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Ирационалан број – Serbian" lang="sr" hreflang="sr" data-title="Ирационалан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Iracionalni_broj" title="Iracionalni broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Iracionalni broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Irrationaaliluku" title="Irrationaaliluku – Finnish" lang="fi" hreflang="fi" data-title="Irrationaaliluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <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 that is not a ratio of integers</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Square_root_of_2_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/240px-Square_root_of_2_triangle.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/360px-Square_root_of_2_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/480px-Square_root_of_2_triangle.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>The number <a href="/wiki/Square_root_of_2" title="Square root of 2"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></a> is irrational.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>irrational numbers</b> (<i><a href="https://en.wiktionary.org/wiki/in-#Etymology_3" class="extiw" title="wikt:in-">in-</a></i> + <i>rational</i>) are all the <a href="/wiki/Real_number" title="Real number">real numbers</a> that are not <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. That is, irrational numbers cannot be expressed as the ratio of two <a href="/wiki/Integer" title="Integer">integers</a>. When the <a href="/wiki/Ratio" title="Ratio">ratio</a> of lengths of two line segments is an irrational number, the line segments are also described as being <i><a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a></i>, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. </p><p>Among irrational numbers are the ratio <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> of a circle's circumference to its diameter, Euler's number <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><i>e</i></a>, the golden ratio <a href="/wiki/Golden_ratio" title="Golden ratio"><i>φ</i></a>, and the <a href="/wiki/Square_root_of_two" class="mw-redirect" title="Square root of two">square root of two</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In fact, all square roots of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, other than of <a href="/wiki/Square_number" title="Square number">perfect squares</a>, are irrational.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Like all real numbers, irrational numbers can be expressed in <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a>, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor <a href="/wiki/Repeating_decimal" title="Repeating decimal">end with a repeating sequence</a>. For example, the decimal representation of <span class="texhtml mvar" style="font-style:italic;">π</span> starts with 3.14159, but no finite number of digits can represent <span class="texhtml mvar" style="font-style:italic;">π</span> exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. </p><p>Irrational numbers can also be expressed as <a href="/wiki/Simple_continued_fraction#Infinite_continued_fractions_and_convergents" title="Simple continued fraction">non-terminating continued fractions</a> (which in some cases are <a href="/wiki/Periodic_continued_fraction" title="Periodic continued fraction">periodic</a>), and in many other ways. </p><p>As a consequence of <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's proof</a> that the real numbers are <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> and the <a href="/wiki/Rational_number" title="Rational number">rationals</a> countable, it follows that <a href="/wiki/Almost_all" title="Almost all">almost all</a> real numbers are irrational.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Set_of_real_numbers_(diagram).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Set_of_real_numbers_%28diagram%29.svg/220px-Set_of_real_numbers_%28diagram%29.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Set_of_real_numbers_%28diagram%29.svg/330px-Set_of_real_numbers_%28diagram%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Set_of_real_numbers_%28diagram%29.svg/440px-Set_of_real_numbers_%28diagram%29.svg.png 2x" data-file-width="500" data-file-height="300" /></a><figcaption>An <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> showing the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of real numbers (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>), which include the rationals (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>), which include the integers (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>), which include the natural numbers (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>). The real numbers also include the irrationals (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>\<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>).</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Ancient_Greece">Ancient Greece</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=2" title="Edit section: Ancient Greece"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first proof of the existence of irrational numbers is usually attributed to a <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagorean</a> (possibly <a href="/wiki/Hippasus" title="Hippasus">Hippasus of Metapontum</a>),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> who probably discovered them while identifying sides of the <a href="/wiki/Pentagram" title="Pentagram">pentagram</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of an <a href="/wiki/Isosceles_right_triangle" class="mw-redirect" title="Isosceles right triangle">isosceles right triangle</a> was indeed <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">commensurable</a> with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows: </p> <ul><li>Start with an isosceles right triangle with side lengths of integers <i>a</i>, <i>b</i>, and <i>c</i>. The ratio of the hypotenuse to a leg is represented by <i>c</i>:<i>b</i>.</li> <li>Assume <i>a</i>, <i>b</i>, and <i>c</i> are in the smallest possible terms (<i>i.e.</i> they have no common factors).</li> <li>By the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: <i>c</i><sup>2</sup> = <i>a</i><sup>2</sup>+<i>b</i><sup>2</sup> = <i>b</i><sup>2</sup>+<i>b</i><sup>2</sup> = 2<i>b</i><sup>2</sup>. (Since the triangle is isosceles, <i>a</i> = <i>b</i>).</li> <li>Since <i>c</i><sup>2</sup> = 2<i>b</i><sup>2</sup>, <i>c</i><sup>2</sup> is divisible by 2, and therefore even.</li> <li>Since <i>c</i><sup>2</sup> is even, <i>c</i> must be even.</li> <li>Since <i>c</i> is even, dividing <i>c</i> by 2 yields an integer. Let <i>y</i> be this integer (<i>c</i> = 2<i>y</i>).</li> <li>Squaring both sides of <i>c</i> = 2<i>y</i> yields <i>c</i><sup>2</sup> = (2<i>y</i>)<sup>2</sup>, or <i>c</i><sup>2</sup> = 4<i>y</i><sup>2</sup>.</li> <li>Substituting 4<i>y</i><sup>2</sup> for <i>c</i><sup>2</sup> in the first equation (<i>c</i><sup>2</sup> = 2<i>b</i><sup>2</sup>) gives us 4<i>y</i><sup>2</sup>= 2<i>b</i><sup>2</sup>.</li> <li>Dividing by 2 yields 2<i>y</i><sup>2</sup> = <i>b</i><sup>2</sup>.</li> <li>Since <i>y</i> is an integer, and 2<i>y</i><sup>2</sup> = <i>b</i><sup>2</sup>, <i>b</i><sup>2</sup> is divisible by 2, and therefore even.</li> <li>Since <i>b</i><sup>2</sup> is even, <i>b</i> must be even.</li> <li>We have just shown that both <i>b</i> and <i>c</i> must be even. Hence they have a common factor of 2. However, this contradicts the assumption that they have no common factors. This contradiction proves that <i>c</i> and <i>b</i> cannot both be integers and thus the existence of a number that cannot be expressed as a ratio of two integers.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <p><a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematicians</a> termed this ratio of incommensurable magnitudes <i>alogos</i>, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.'<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory. </p><p>The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This was brought to light by <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a>, who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for "whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects",<sup id="cite_ref-Kline_1990,_p._34_8-0" class="reference"><a href="#cite_note-Kline_1990,_p._34-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> but Zeno found that in fact "[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear... .[Q]uantities are, in other words, continuous".<sup id="cite_ref-Kline_1990,_p._34_8-1" class="reference"><a href="#cite_note-Kline_1990,_p._34-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> What this means is that contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily be <a href="/wiki/Infinity" title="Infinity">infinite</a>. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating <a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">four paradoxes</a>, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore, further investigation had to occur. </p><p>The next step was taken by <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. "Eudoxus' theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios".<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of <i>x</i><sup>2</sup> and <i>x</i><sup>3</sup> as <i>x</i> squared and <i>x</i> cubed instead of <i>x</i> to the second power and <i>x</i> to the third power. Also crucial to Zeno's work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this necessity, Eudoxus developed his <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, a kind of <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a> that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof".<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This method of exhaustion is the first step in the creation of calculus. </p><p><a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus of Cyrene</a> proved the irrationality of the <a href="/wiki/Nth_root" title="Nth root">surds</a> of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="India">India</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=3" title="Edit section: India"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the <a href="/wiki/Vedic_period" title="Vedic period">Vedic period</a> in India. There are references to such calculations in the <i><a href="/wiki/Samhita" title="Samhita">Samhitas</a></i>, <i><a href="/wiki/Brahmana" title="Brahmana">Brahmanas</a></i>, and the <i><a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a></i> (800 BC or earlier).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>It is suggested that the concept of irrationality was implicitly accepted by <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematicians</a> since the 7th century BC, when <a href="/wiki/Manava" title="Manava">Manava</a> (c. 750 – 690 BC) believed that the <a href="/wiki/Square_root" title="Square root">square roots</a> of numbers such as 2 and 61 could not be exactly determined.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Historian <a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Carl Benjamin Boyer</a>, however, writes that "such claims are not well substantiated and unlikely to be true".<sup id="cite_ref-Boyer_208_16-0" class="reference"><a href="#cite_note-Boyer_208-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Mathematicians like <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (in 628 AD) and <a href="/wiki/Bh%C4%81skara_I" title="Bhāskara I">Bhāskara I</a> (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a> evaluated some of these formulas and critiqued them, identifying their limitations. </p><p>During the 14th to 16th centuries, <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> and the <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a> discovered the <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> for several irrational numbers such as <i><a href="/wiki/Pi" title="Pi">π</a></i> and certain irrational values of <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>. <a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a> provided proofs for these infinite series in the <i><a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a></i>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Islamic_World">Islamic World</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=4" title="Edit section: Islamic World"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <a href="/wiki/Middle_Ages" title="Middle Ages">Middle Ages</a>, the development of <a href="/wiki/Algebra" title="Algebra">algebra</a> by <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Muslim mathematicians</a> allowed irrational numbers to be treated as <i>algebraic objects</i>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Middle Eastern mathematicians also merged the concepts of "<a href="/wiki/Number" title="Number">number</a>" and "<a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>" into a more general idea of <a href="/wiki/Real_number" title="Real number">real numbers</a>, criticized Euclid's idea of <a href="/wiki/Ratio" title="Ratio">ratios</a>, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.<sup id="cite_ref-Matvievskaya_20-0" class="reference"><a href="#cite_note-Matvievskaya-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In his commentary on Book 10 of the <i>Elements</i>, the <a href="/wiki/Persian_people" class="mw-redirect" title="Persian people">Persian</a> mathematician <a href="/wiki/Al-Mahani" title="Al-Mahani">Al-Mahani</a> (d. 874/884) examined and classified <a href="/wiki/Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrationals</a> and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:<sup id="cite_ref-Matvievskaya_20-1" class="reference"><a href="#cite_note-Matvievskaya-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes <i>etc.</i>"</p></blockquote> <p>In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and <a href="/wiki/Cube_root" title="Cube root">cube roots</a> as irrational magnitudes. He also introduced an <a href="/wiki/Arithmetic" title="Arithmetic">arithmetical</a> approach to the concept of irrationality, as he attributes the following to irrational magnitudes:<sup id="cite_ref-Matvievskaya_20-2" class="reference"><a href="#cite_note-Matvievskaya-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."</p></blockquote> <p>The <a href="/wiki/Egypt" title="Egypt">Egyptian</a> mathematician <a href="/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81_ibn_Aslam" class="mw-redirect" title="Abū Kāmil Shujā ibn Aslam">Abū Kāmil Shujā ibn Aslam</a> (c. 850 – 930) was the first to accept irrational numbers as solutions to <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a> or as <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in an <a href="/wiki/Equation" title="Equation">equation</a> in the form of square roots and <a href="/wiki/Nth_root" title="Nth root">fourth roots</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In the 10th century, the <a href="/wiki/Iraq" title="Iraq">Iraqi</a> mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.<sup id="cite_ref-Matvievskaya_20-3" class="reference"><a href="#cite_note-Matvievskaya-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Many of these concepts were eventually accepted by European mathematicians sometime after the <a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a>. <a href="/wiki/Al-Hass%C4%81r" class="mw-redirect" title="Al-Hassār">Al-Hassār</a>, a Moroccan mathematician from <a href="/wiki/Fes" class="mw-redirect" title="Fes">Fez</a> specializing in <a href="/wiki/Islamic_inheritance_jurisprudence" title="Islamic inheritance jurisprudence">Islamic inheritance jurisprudence</a> during the 12th century, first mentions the use of a fractional bar, where <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerators</a> and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3\quad 1}{5\quad 3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mspace width="1em" /> <mn>1</mn> </mrow> <mrow> <mn>5</mn> <mspace width="1em" /> <mn>3</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\quad 1}{5\quad 3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078d004ad0c501c44d3ff1d2360ade80bb2fad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.484ex; height:5.176ex;" alt="{\displaystyle {\frac {3\quad 1}{5\quad 3}}}"></span>."<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This same fractional notation appears soon after in the work of <a href="/wiki/Leonardo_Fibonacci" class="mw-redirect" title="Leonardo Fibonacci">Leonardo Fibonacci</a> in the 13th century.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Modern_period">Modern period</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=5" title="Edit section: Modern period"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 17th century saw <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary numbers</a> become a powerful tool in the hands of <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>, and especially of <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. The completion of the theory of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> in the 19th century entailed the differentiation of irrationals into algebraic and <a href="/wiki/Transcendental_numbers" class="mw-redirect" title="Transcendental numbers">transcendental numbers</a>, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since <a href="/wiki/Euclid" title="Euclid">Euclid</a>. The year 1872 saw the publication of the theories of <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> (by his pupil Ernst Kossak), <a href="/wiki/Eduard_Heine" title="Eduard Heine">Eduard Heine</a> (<i><a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Crelle's Journal</a></i>, 74), <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> (Annalen, 5), and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by <a href="/wiki/Salvatore_Pincherle" title="Salvatore Pincherle">Salvatore Pincherle</a> in 1880,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by <a href="/wiki/Paul_Tannery" title="Paul Tannery">Paul Tannery</a> (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a <a href="/wiki/Dedekind_cut" title="Dedekind cut">cut (Schnitt)</a> in the system of all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, separating them into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> (Crelle, 101), and <a href="/wiki/Charles_M%C3%A9ray" title="Charles Méray">Charles Méray</a>. </p><p><a href="/wiki/Continued_fraction" title="Continued fraction">Continued fractions</a>, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. </p><p><a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> proved (1761) that π cannot be rational, and that <i>e</i><sup><i>n</i></sup> is irrational if <i>n</i> is rational (unless <i>n</i> = 0).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> (1794), after introducing the <a href="/wiki/Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a>, provided a proof to show that π<sup>2</sup> is irrational, whence it follows immediately that π is irrational also. The existence of <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a> was first established by Liouville (1844, 1851). Later, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> (1873) proved their existence by a <a href="/wiki/Georg_Cantor%27s_first_set_theory_article" class="mw-redirect" title="Georg Cantor's first set theory article">different method</a>, which showed that every interval in the reals contains transcendental numbers. <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a> (1873) first proved <i>e</i> transcendental, and <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Ferdinand von Lindemann</a> (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> (1893), and was finally made elementary by <a href="/wiki/Adolf_Hurwitz" title="Adolf Hurwitz">Adolf Hurwitz</a><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2018)">citation needed</span></a></i>]</sup> and <a href="/wiki/Paul_Gordan" title="Paul Gordan">Paul Gordan</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Irrational_number" title="Special:EditPage/Irrational number">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Irrational+number%22">"Irrational number"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Irrational+number%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Irrational+number%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Irrational+number%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Irrational+number%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Irrational+number%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Square_roots">Square roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=7" title="Edit section: Square roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> was likely the first number proved irrational.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> is another famous quadratic irrational number. The square roots of all natural numbers that are not <a href="/wiki/Perfect_squares" class="mw-redirect" title="Perfect squares">perfect squares</a> are irrational and a proof may be found in <a href="/wiki/Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrationals</a>. </p> <div class="mw-heading mw-heading3"><h3 id="General_roots">General roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=8" title="Edit section: General roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The proof for the irrationality of the square root of two can be generalized using the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. This asserts that every integer has a <a href="/wiki/Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a> into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in <a href="/wiki/Lowest_terms" class="mw-redirect" title="Lowest terms">lowest terms</a> there must be a <a href="/wiki/Prime_number" title="Prime number">prime</a> in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact <span class="texhtml mvar" style="font-style:italic;">k</span>th power of another integer, then that first integer's <a href="/wiki/Nth_root" title="Nth root"><span class="texhtml mvar" style="font-style:italic;">k</span>th root</a> is irrational. </p> <div class="mw-heading mw-heading3"><h3 id="Logarithms">Logarithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=9" title="Edit section: Logarithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Perhaps the numbers most easy to prove irrational are certain <a href="/wiki/Logarithm" title="Logarithm">logarithms</a>. Here is a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> that log<sub>2</sub> 3 is irrational (log<sub>2</sub> 3 ≈ 1.58 > 0). </p><p>Assume log<sub>2</sub> 3 is rational. For some positive integers <i>m</i> and <i>n</i>, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}3={\frac {m}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}3={\frac {m}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590b1df5ab889f2301ad8f0e2850c173b0335047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.197ex; height:4.676ex;" alt="{\displaystyle \log _{2}3={\frac {m}{n}}.}"></span></dd></dl> <p>It follows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m/n}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m/n}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010ee877bce77a92d7db216fa247743e660e0047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.907ex; height:2.843ex;" alt="{\displaystyle 2^{m/n}=3}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{m/n})^{n}=3^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{m/n})^{n}=3^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f4940b4c362bf90132eb134fbb6122c32599b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.153ex; height:3.343ex;" alt="{\displaystyle (2^{m/n})^{n}=3^{n}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m}=3^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}=3^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e108604a895d0d63b4f3c62f0b48e85ce0d3a705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.964ex; height:2.343ex;" alt="{\displaystyle 2^{m}=3^{n}.}"></span></dd></dl> <p>The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log<sub>2</sub> 3 is rational (and so expressible as a quotient of integers <i>m</i>/<i>n</i> with <i>n</i> ≠ 0). The contradiction means that this assumption must be false, i.e. log<sub>2</sub> 3 is irrational, and can never be expressed as a quotient of integers <i>m</i>/<i>n</i> with <i>n</i> ≠ 0. </p><p>Cases such as log<sub>10</sub> 2 can be treated similarly. </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=10" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An irrational number may be <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic</a>, that is a real <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with integer coefficients. Those that are not algebraic are <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic">Algebraic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=11" title="Edit section: Algebraic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The real <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a> are the real solutions of polynomial equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b854d2390d5f3a5d3598e90cd21799def99c5ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:47.226ex; height:3.176ex;" alt="{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0\;,}"></span></dd></dl> <p>where the coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> are integers and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf50204372c4d2e24c37aace0035de48b877973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.709ex; height:2.676ex;" alt="{\displaystyle a_{n}\neq 0}"></span>. An example of an irrational algebraic number is <i>x</i><sub>0</sub> = (2<sup>1/2</sup> + 1)<sup>1/3</sup>. It is clearly algebraic since it is the root of an integer polynomial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{3}-1)^{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{3}-1)^{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd66c1a5404f6d7579355bbe48d484b038af1f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.511ex; height:3.176ex;" alt="{\displaystyle (x^{3}-1)^{2}=2}"></span>, which is equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{6}-2x^{3}-1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{6}-2x^{3}-1)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec4c1d5deed02f853d78182361b932213706ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.844ex; height:3.176ex;" alt="{\displaystyle (x^{6}-2x^{3}-1)=0}"></span>. This polynomial has no rational roots, since the <a href="/wiki/Rational_root_theorem" title="Rational root theorem">rational root theorem</a> shows that the only possibilities are ±1, but <i>x</i><sub>0</sub> is greater than 1. So <i>x</i><sub>0</sub> is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials. </p> <div class="mw-heading mw-heading3"><h3 id="Transcendental">Transcendental</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=12" title="Edit section: Transcendental"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Almost_all" title="Almost all">Almost all</a> irrational numbers are <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>. Examples are <i>e</i><sup> <i>r</i></sup> and π<sup> <i>r</i></sup>, which are transcendental for all nonzero rational <i>r.</i> </p><p>Because the algebraic numbers form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">subfield</a> of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3<span class="texhtml mvar" style="font-style:italic;">π</span> + 2, <span class="texhtml mvar" style="font-style:italic;">π</span> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> and <i>e</i><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span> are irrational (and even transcendental). </p> <div class="mw-heading mw-heading2"><h2 id="Decimal_expansions">Decimal expansions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=13" title="Edit section: Decimal expansions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there is not a finite number of nonzero digits), unlike any rational number. The same is true for <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a>, <a href="/wiki/Octal" title="Octal">octal</a> or <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> expansions, and in general for expansions in every <a href="/wiki/Positional_notation" title="Positional notation">positional</a> <a href="/wiki/Numeral_system" title="Numeral system">notation</a> with <a href="/wiki/Natural_number" title="Natural number">natural</a> bases. </p><p>To show this, suppose we divide integers <i>n</i> by <i>m</i> (where <i>m</i> is nonzero). When <a href="/wiki/Long_division" title="Long division">long division</a> is applied to the division of <i>n</i> by <i>m</i>, there can never be a <a href="/wiki/Remainder" title="Remainder">remainder</a> greater than or equal to <i>m</i>. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most <i>m</i> − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats. </p><p>Conversely, suppose we are faced with a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a>, we can prove that it is a fraction of two integers. For example, consider: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=0.7\,162\,162\,162\,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>0.7</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=0.7\,162\,162\,162\,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ffe004b8de2fb139a6ff54b6b9633a55cc106d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.934ex; height:2.176ex;" alt="{\displaystyle A=0.7\,162\,162\,162\,\ldots }"></span></dd></dl> <p>Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10A=7.162\,162\,162\,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>A</mi> <mo>=</mo> <mn>7.162</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10A=7.162\,162\,162\,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca13e232deb4235ac6752ca6056ab36acec9779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.71ex; height:2.176ex;" alt="{\displaystyle 10A=7.162\,162\,162\,\ldots }"></span></dd></dl> <p>Now we multiply this equation by 10<sup><i>r</i></sup> where <i>r</i> is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10<sup>3</sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10,000A=7\,162.162\,162\,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo>,</mo> <mn>000</mn> <mi>A</mi> <mo>=</mo> <mn>7</mn> <mspace width="thinmathspace" /> <mn>162.162</mn> <mspace width="thinmathspace" /> <mn>162</mn> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10,000A=7\,162.162\,162\,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b8e0f07d63e789770e0ff1fc7012c2b6068322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.231ex; height:2.509ex;" alt="{\displaystyle 10,000A=7\,162.162\,162\,\ldots }"></span></dd></dl> <p>The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000<i>A</i> matches the tail end of 10<i>A</i> exactly. Here, both 10,000<i>A</i> and 10<i>A</i> have <span style="white-space:nowrap">.162<span style="margin-left:0.25em">162</span><span style="margin-left:0.25em">162</span><span style="margin-left:0.25em">...</span></span> after the decimal point. </p><p>Therefore, when we subtract the 10<i>A</i> equation from the 10,000<i>A</i> equation, the tail end of 10<i>A</i> cancels out the tail end of 10,000<i>A</i> leaving us with: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9990A=7155.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9990</mn> <mi>A</mi> <mo>=</mo> <mn>7155.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9990A=7155.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40427e8a0951e78459e5af9e55e1565075b9e6bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.788ex; height:2.176ex;" alt="{\displaystyle 9990A=7155.}"></span></dd></dl> <p>Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {7155}{9990}}={\frac {53}{74}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7155</mn> <mn>9990</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>53</mn> <mn>74</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {7155}{9990}}={\frac {53}{74}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032c054c4322be50181fde68615431554ee08e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.587ex; height:5.343ex;" alt="{\displaystyle A={\frac {7155}{9990}}={\frac {53}{74}}}"></span></dd></dl> <p>is a ratio of integers and therefore a rational number. </p> <div class="mw-heading mw-heading2"><h2 id="Irrational_powers">Irrational powers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=14" title="Edit section: Irrational powers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dov Jarden gave a simple non-<a href="/wiki/Constructive_proof" title="Constructive proof">constructive proof</a> that there exist two irrational numbers <i>a</i> and <i>b</i>, such that <i>a</i><sup><i>b</i></sup> is rational:<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>Consider <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup>; if this is rational, then take <i>a</i> = <i>b</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. Otherwise, take <i>a</i> to be the irrational number <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup> and <i>b</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. Then <i>a</i><sup><i>b</i></sup> = (<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup>)<sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>·<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup>2</sup> = 2, which is rational.</dd></dl> <p>Although the above argument does not decide between the two cases, the <a href="/wiki/Gelfond%E2%80%93Schneider_theorem" title="Gelfond–Schneider theorem">Gelfond–Schneider theorem</a> shows that <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span><sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup> is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>, hence irrational. This theorem states that if <i>a</i> and <i>b</i> are both <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, and <i>a</i> is not equal to 0 or 1, and <i>b</i> is not a rational number, then any value of <i>a</i><sup><i>b</i></sup> is a transcendental number (there can be more than one value if <a href="/wiki/Exponentiation#nth_roots_of_a_complex_number" title="Exponentiation">complex number exponentiation</a> is used). </p><p>An example that provides a simple constructive proof is<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt {2}}\right)^{\log _{\sqrt {2}}3}=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {2}}\right)^{\log _{\sqrt {2}}3}=3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691a70feb6ed00ab54c16f3c5ebc4418a70cae3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.621ex; height:3.843ex;" alt="{\displaystyle \left({\sqrt {2}}\right)^{\log _{\sqrt {2}}3}=3.}"></span></dd></dl> <p>The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{\sqrt {2}}3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\sqrt {2}}3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e03525c6cba46bb434faf9e4d12f05efbea3d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.944ex; height:3.009ex;" alt="{\displaystyle \log _{\sqrt {2}}3}"></span>, is irrational. This is so because, by the formula relating logarithms with different bases, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{\sqrt {2}}3={\frac {\log _{2}3}{\log _{2}{\sqrt {2}}}}={\frac {\log _{2}3}{1/2}}=2\log _{2}3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mrow> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\sqrt {2}}3={\frac {\log _{2}3}{\log _{2}{\sqrt {2}}}}={\frac {\log _{2}3}{1/2}}=2\log _{2}3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0220e3713097d612ff9c09bbb75afa5fbd3277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.124ex; height:6.676ex;" alt="{\displaystyle \log _{\sqrt {2}}3={\frac {\log _{2}3}{\log _{2}{\sqrt {2}}}}={\frac {\log _{2}3}{1/2}}=2\log _{2}3}"></span></dd></dl> <p>which we can assume, for the sake of establishing a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">contradiction</a>, equals a ratio <i>m/n</i> of positive integers. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}3=m/2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}3=m/2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71279224692143f506d16885471746a5c991fe53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.434ex; height:2.843ex;" alt="{\displaystyle \log _{2}3=m/2n}"></span> hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\log _{2}3}=2^{m/2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\log _{2}3}=2^{m/2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07e0fa7cf01c98025273df02b23d43ca5617bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.103ex; height:2.843ex;" alt="{\displaystyle 2^{\log _{2}3}=2^{m/2n}}"></span> hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3=2^{m/2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3=2^{m/2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fab808028a1fefa00694eeb7883eefa51a69510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.729ex; height:2.843ex;" alt="{\displaystyle 3=2^{m/2n}}"></span> hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2n}=2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2n}=2^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3532ce369b819b570afbe183d77f2aed8b38d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.139ex; height:2.676ex;" alt="{\displaystyle 3^{2n}=2^{m}}"></span>, which is a contradictory pair of prime factorizations and hence violates the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a> (unique prime factorization). </p><p>A stronger result is the following:<sup id="cite_ref-Marshall_31-0" class="reference"><a href="#cite_note-Marshall-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Every rational number in the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((1/e)^{1/e},\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> </mrow> </msup> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((1/e)^{1/e},\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777fe7bcf392fdf1445b84e7e84185032ebedca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.027ex; height:3.343ex;" alt="{\displaystyle ((1/e)^{1/e},\infty )}"></span> can be written either as <i>a</i><sup><i>a</i></sup> for some irrational number <i>a</i> or as <i>n</i><sup><i>n</i></sup> for some natural number <i>n</i>. Similarly,<sup id="cite_ref-Marshall_31-1" class="reference"><a href="#cite_note-Marshall-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> every positive rational number can be written either as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{a^{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{a^{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc5467ac61c4da02930e211436a5a75e1fd7881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.202ex; height:2.676ex;" alt="{\displaystyle a^{a^{a}}}"></span> for some irrational number <i>a</i> or as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{n^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{n^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/621e862c819a7e38ee29474819c0a5a39fdb3958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.578ex; height:2.676ex;" alt="{\displaystyle n^{n^{n}}}"></span> for some natural number <i>n</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Open_questions">Open questions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=15" title="Edit section: Open questions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Various combinations of <span class="texhtml"><i>e</i></span>, <span class="texhtml"><i>π</i></span> and <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a> (such as <span class="texhtml"><i>e</i> + <i>π</i></span><i>, <span class="texhtml mvar" style="font-style:italic;">eπ</span></i>, <span class="texhtml"><i>e</i><sup><i>e</i></sup></span>, <span class="texhtml"><i>π</i><sup><i>e</i></sup></span>, <i><span class="texhtml mvar" style="font-style:italic;">π</span><sup><span class="texhtml mvar" style="font-style:italic;">π</span></sup></i>, <span class="texhtml">ln <i>π</i></span>) are not known to be irrational, in part because <span class="texhtml"><i>e</i></span> and <span class="texhtml"><i>π</i></span> are not known to be <a href="/wiki/Algebraic_independence" title="Algebraic independence">algebraically independent</a>. <a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a> would imply that all of the above numbers are irrational and even transcendental.<sup id="cite_ref-:12_32-0" class="reference"><a href="#cite_note-:12-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li> <li>The question about the irrationality of <a href="/wiki/Euler%27s_constant" title="Euler's constant">Euler's constant</a> <i><span class="texhtml mvar" style="font-style:italic;">γ</span></i> is a long standing <a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">open problem</a> in <a href="/wiki/Number_theory" title="Number theory">number theory</a>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li> <li>Other important numbers which are not known to be irrational include <a href="/wiki/Particular_values_of_the_Riemann_zeta_function#Odd_positive_integers" title="Particular values of the Riemann zeta function">odd zeta constants</a> <span class="texhtml"><i>ζ</i>(5), <i>ζ</i>(7), <i>ζ</i>(9), ...</span> for <span class="texhtml">n > 3</span> and <a href="/wiki/Catalan%27s_constant" title="Catalan's constant">Catalan's constant</a> <span class="texhtml"><i>β</i>(2)</span>.<sup id="cite_ref-:4_34-0" class="reference"><a href="#cite_note-:4-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="In_constructive_mathematics">In constructive mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=16" title="Edit section: In constructive mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructive mathematics</a>, <a href="/wiki/Excluded_middle" class="mw-redirect" title="Excluded middle">excluded middle</a> is not valid, so it is not true that every real number is rational or irrational. Thus, the notion of an irrational number bifurcates into multiple distinct notions. One could take the traditional definition of an irrational number as a real number that is not rational.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> However, there is a second definition of an irrational number used in constructive mathematics, that a real number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is an irrational number if it is <a href="/wiki/Apartness_relation" title="Apartness relation">apart</a> from every rational number, or equivalently, if the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert r-q\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>r</mi> <mo>−</mo> <mi>q</mi> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert r-q\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1809dd5ece3247ac662aae272887ada2e561ea80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.252ex; height:2.843ex;" alt="{\displaystyle \vert r-q\vert }"></span> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and every rational number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> is positive. This definition is stronger than the traditional definition of an irrational number. This second definition is used in <a href="/wiki/Errett_Bishop" title="Errett Bishop">Errett Bishop</a>'s <a href="/wiki/Square_root_of_2#Constructive_proof" title="Square root of 2">proof that the square root of 2 is irrational</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Set_of_all_irrationals">Set of all irrationals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=17" title="Edit section: Set of all irrationals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the reals form an <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> set, of which the rationals are a <a href="/wiki/Countable_set" title="Countable set">countable</a> subset, the complementary set of irrationals is uncountable. </p><p>Under the usual (<a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean</a>) distance function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\vert x-y\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\vert x-y\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f2c910a5f4d4ab774a4671c12178d63eb66b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle d(x,y)=\vert x-y\vert }"></span>, the real numbers are a <a href="/wiki/Metric_space" title="Metric space">metric space</a> and hence also a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed, the induced metric is not <a href="/wiki/Complete_(topology)" class="mw-redirect" title="Complete (topology)">complete</a>. Being a <a href="/wiki/G-delta_set" class="mw-redirect" title="G-delta set">G-delta set</a>—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is <a href="/wiki/Completely_metrizable" class="mw-redirect" title="Completely metrizable">completely metrizable</a>: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable. </p><p>Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen groups so the space is <a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">zero-dimensional</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Brjuno_number" title="Brjuno number">Brjuno number</a></li> <li><a href="/wiki/Computable_number" title="Computable number">Computable number</a></li> <li><a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a></li> <li><a href="/wiki/Irrationality_measure" title="Irrationality measure">Irrationality measure</a></li> <li><a href="/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">Proof that <span class="texhtml mvar" style="font-style:italic;">e</span> is irrational</a></li> <li><a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">Proof that <span class="texhtml mvar" style="font-style:italic;">π</span> is irrational</a></li> <li><a href="/wiki/Square_root_of_3" title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Trigonometric_number" class="mw-redirect" title="Trigonometric number">Trigonometric number</a></li></ul> <table style="margin:2em; border:2px solid silver; font-size:95%; border-collapse:collapse"> <tbody><tr> <td> <table style="margin:4px; border:2px solid silver"> <tbody><tr> <td> <table style="margin:1em"> <caption><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number systems</a> </caption> <tbody><tr> <td><a href="/wiki/Complex_number" title="Complex number">Complex</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {C} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Real_number" title="Real number">Real</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {R} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Rational_number" title="Rational number">Rational</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Integer" title="Integer">Integer</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {Z} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Natural_number" title="Natural number">Natural</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>: 0 </td></tr> <tr> <td><a href="/wiki/One" class="mw-redirect" title="One">One</a>: 1 </td></tr> <tr> <td><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> </td></tr> <tr> <td><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Fraction" title="Fraction">Fraction</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a> </td></tr> <tr> <td><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a> </td></tr> <tr> <td><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> </td> <td> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a class="mw-selflink selflink">Irrational</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic irrational</a> </td></tr> <tr> <td><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a> </td></tr> <tr> <td><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://sprott.physics.wisc.edu/Pickover/trans.html">The 15 Most Famous Transcendental Numbers</a>. by <a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a>. URL retrieved 24 October 2007.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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="CITEREFJackson2011" class="citation journal cs1">Jackson, Terence (2011-07-01). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/9542-irrational-square-roots-of-natural-numbers-a-geometrical-approach/6B9D8EBFDCC016013D303AA78973429F">"95.42 Irrational square roots of natural numbers — a geometrical approach"</a>. <i>The Mathematical Gazette</i>. <b>95</b> (533): 327–330. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0025557200003193">10.1017/S0025557200003193</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/0025-5572">0025-5572</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:123995083">123995083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=95.42+Irrational+square+roots+of+natural+numbers+%E2%80%94+a+geometrical+approach&rft.volume=95&rft.issue=533&rft.pages=327-330&rft.date=2011-07-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123995083%23id-name%3DS2CID&rft.issn=0025-5572&rft_id=info%3Adoi%2F10.1017%2FS0025557200003193&rft.aulast=Jackson&rft.aufirst=Terence&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fjournals%2Fmathematical-gazette%2Farticle%2Fabs%2F9542-irrational-square-roots-of-natural-numbers-a-geometrical-approach%2F6B9D8EBFDCC016013D303AA78973429F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1955" class="citation book cs1"><a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor, Georg</a> (1955) [1915]. <a href="/wiki/Philip_Jourdain" title="Philip Jourdain">Philip Jourdain</a> (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/contributionstot003626mbp"><i>Contributions to the Founding of the Theory of Transfinite Numbers</i></a>. New York: Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60045-1" title="Special:BookSources/978-0-486-60045-1"><bdi>978-0-486-60045-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contributions+to+the+Founding+of+the+Theory+of+Transfinite+Numbers&rft.place=New+York&rft.pub=Dover&rft.date=1955&rft.isbn=978-0-486-60045-1&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcontributionstot003626mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurt_Von_Fritz1945" class="citation journal cs1"><a href="/wiki/Kurt_von_Fritz" title="Kurt von Fritz">Kurt Von Fritz</a> (1945). 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"The Pentagram and the Discovery of an Irrational Number". <i>The Two-Year College Mathematics Journal</i>. <b>11</b> (5): 312–316. <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%2F3026893">10.2307/3026893</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/3026893">3026893</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:115390951">115390951</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=The+Pentagram+and+the+Discovery+of+an+Irrational+Number&rft.volume=11&rft.issue=5&rft.pages=312-316&rft.date=1980&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115390951%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3026893%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3026893&rft.au=James+R.+Choike&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, M.</a> (1990). <i>Mathematical Thought from Ancient to Modern Times</i>, Vol. 1. New York: Oxford University Press (original work published 1972), p. 33.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Kline 1990, p. 32.</span> </li> <li id="cite_note-Kline_1990,_p._34-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kline_1990,_p._34_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kline_1990,_p._34_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Kline 1990, p. 34.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Kline 1990, p. 48.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Kline 1990, p. 49.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharles_H._Edwards1982" class="citation book cs1">Charles H. 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"Theodorus' Irrationality Proofs". <i>Mathematics Magazine</i>. <b>49</b> (4): 201–203. <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.1976.11976579">10.1080/0025570X.1976.11976579</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/2690123">2690123</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:124565880">124565880</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=Theodorus%27+Irrationality+Proofs&rft.volume=49&rft.issue=4&rft.pages=201-203&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124565880%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690123%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F0025570X.1976.11976579&rft.au=Robert+L.+McCabe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBag1990" class="citation journal cs1">Bag, Amulya Kumar (1990). <a rel="nofollow" class="external text" href="http://repository.ias.ac.in/74671/">"Ritual Geometry in India and its Parallelism in other Culture Areas"</a>. <i>Indian Journal of History of Science</i>. <b>25</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Journal+of+History+of+Science&rft.atitle=Ritual+Geometry+in+India+and+its+Parallelism+in+other+Culture+Areas&rft.volume=25&rft.date=1990&rft.aulast=Bag&rft.aufirst=Amulya+Kumar&rft_id=http%3A%2F%2Frepository.ias.ac.in%2F74671%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSelinD'Ambrosio2000" class="citation book cs1"><a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a>; D'Ambrosio, Ubiratan, eds. (2000). <i>Mathematics Across Cultures: The History of Non-western Mathematics</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0260-2" title="Special:BookSources/1-4020-0260-2"><bdi>1-4020-0260-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Across+Cultures%3A+The+History+of+Non-western+Mathematics&rft.pub=Springer&rft.date=2000&rft.isbn=1-4020-0260-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span>.</span> </li> <li id="cite_note-Boyer_208-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer_208_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1991" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer</a> (1991). 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The case for early Hindu awareness of incommensurable magnitudes is rendered most unlikely by the lack of evidence that Indian mathematicians of that period had come to grips with fundamental concepts.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=China+and+India&rft.btitle=A+History+of+Mathematics&rft.pages=208&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft_id=info%3Aoclcnum%2F414892&rft.isbn=0471093742&rft.au=Boyer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDattaSingh1993" class="citation journal cs1">Datta, Bibhutibhusan; Singh, Awadhesh Narayan (1993). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181003062219/https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol28_3_2_BDatta.pdf">"Surds in Hindu mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>Indian Journal of History of Science</i>. <b>28</b> (3): 253–264. 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Retrieved <span class="nowrap">18 September</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Journal+of+History+of+Science&rft.atitle=Surds+in+Hindu+mathematics&rft.volume=28&rft.issue=3&rft.pages=253-264&rft.date=1993&rft.aulast=Datta&rft.aufirst=Bibhutibhusan&rft.au=Singh%2C+Awadhesh+Narayan&rft_id=https%3A%2F%2Finsa.nic.in%2Fwritereaddata%2FUpLoadedFiles%2FIJHS%2FVol28_3_2_BDatta.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1995" class="citation journal cs1"><a href="/wiki/Victor_J._Katz" title="Victor J. Katz">Katz, V. J.</a> (1995). "Ideas of Calculus in Islam and India". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>63</b> (3): 163–174. <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%2F2691411">10.2307/2691411</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/2691411">2691411</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=Ideas+of+Calculus+in+Islam+and+India&rft.volume=63&rft.issue=3&rft.pages=163-174&rft.date=1995&rft_id=info%3Adoi%2F10.2307%2F2691411&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2691411%23id-name%3DJSTOR&rft.aulast=Katz&rft.aufirst=V.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson1999" class="citation cs1">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> (1999). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html">"Arabic mathematics: forgotten brilliance?"</a>. <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>. <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Arabic+mathematics%3A+forgotten+brilliance%3F&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.date=1999&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FArabic_mathematics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span>.</span> </li> <li id="cite_note-Matvievskaya-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Matvievskaya_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Matvievskaya_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Matvievskaya_20-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Matvievskaya_20-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatvievskaya1987" class="citation journal cs1"><a href="/w/index.php?title=Galina_Matvievskaya&action=edit&redlink=1" class="new" title="Galina Matvievskaya (page does not exist)">Matvievskaya, Galina</a> (1987). "The theory of quadratic irrationals in medieval Oriental mathematics". <i><a href="/wiki/New_York_Academy_of_Sciences" title="New York Academy of Sciences">Annals of the New York Academy of Sciences</a></i>. <b>500</b> (1): 253–277. <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/1987NYASA.500..253M">1987NYASA.500..253M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1749-6632.1987.tb37206.x">10.1111/j.1749-6632.1987.tb37206.x</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:121416910">121416910</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+New+York+Academy+of+Sciences&rft.atitle=The+theory+of+quadratic+irrationals+in+medieval+Oriental+mathematics&rft.volume=500&rft.issue=1&rft.pages=253-277&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121416910%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fj.1749-6632.1987.tb37206.x&rft_id=info%3Abibcode%2F1987NYASA.500..253M&rft.aulast=Matvievskaya&rft.aufirst=Galina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span> See in particular pp. 254 & 259–260.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Jacques Sesiano, "Islamic mathematics", p. 148, in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSelinD'Ambrosio2000" class="citation book cs1">Selin, Helaine; D'Ambrosio, Ubiratan (2000). <i>Mathematics Across Cultures: The History of Non-western Mathematics</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0260-2" title="Special:BookSources/1-4020-0260-2"><bdi>1-4020-0260-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Across+Cultures%3A+The+History+of+Non-western+Mathematics&rft.pub=Springer&rft.date=2000&rft.isbn=1-4020-0260-2&rft.aulast=Selin&rft.aufirst=Helaine&rft.au=D%27Ambrosio%2C+Ubiratan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1928" class="citation book cs1">Cajori, Florian (1928). <i>A History of Mathematical Notations (Vol.1)</i>. La Salle, Illinois: The Open Court Publishing Company.</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+%28Vol.1%29&rft.place=La+Salle%2C+Illinois&rft.pub=The+Open+Court+Publishing+Company&rft.date=1928&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span> pg. 269.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">(<a href="#CITEREFCajori1928">Cajori 1928</a>, pg.89)</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalvatore_Pincherle1880" class="citation journal cs1">Salvatore Pincherle (1880). <a rel="nofollow" class="external text" href="https://archive.org/stream/giornaledimatem07unkngoog#page/n198">"Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principii del prof. C. Weierstrass"</a>. <i>Giornale di Matematiche</i>: 178–254, 317–320.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Giornale+di+Matematiche&rft.atitle=Saggio+di+una+introduzione+alla+teoria+delle+funzioni+analitiche+secondo+i+principii+del+prof.+C.+Weierstrass&rft.pages=178-254%2C+317-320&rft.date=1880&rft.au=Salvatore+Pincherle&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fgiornaledimatem07unkngoog%23page%2Fn198&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLambert1761" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Lambert, J. H.</a> (1761). <a rel="nofollow" class="external text" href="http://www.kuttaka.org/~JHL/L1768b.pdf">"Mémoire sur quelques propriétés remarquables des quantités transcendentes, circulaires et logarithmiques"</a> <span class="cs1-format">(PDF)</span>. <i>Mémoires de l'Académie royale des sciences de Berlin</i> (in French): 265–322. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160428083325/http://www.kuttaka.org/~JHL/L1768b.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2016-04-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+royale+des+sciences+de+Berlin&rft.atitle=M%C3%A9moire+sur+quelques+propri%C3%A9t%C3%A9s+remarquables+des+quantit%C3%A9s+transcendentes%2C+circulaires+et+logarithmiques&rft.pages=265-322&rft.date=1761&rft.aulast=Lambert&rft.aufirst=J.+H.&rft_id=http%3A%2F%2Fwww.kuttaka.org%2F~JHL%2FL1768b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGordan1893" class="citation journal cs1">Gordan, Paul (1893). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428218">"Transcendenz von <i>e</i> und π"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>43</b> (2–3). 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(2001), "The story of the discovery of incommensurability, revisited", <i>Neusis</i> (10): 45–61, <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=1891736">1891736</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neusis&rft.atitle=The+story+of+the+discovery+of+incommensurability%2C+revisited&rft.issue=10&rft.pages=45-61&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1891736%23id-name%3DMR&rft.aulast=Fowler&rft.aufirst=David+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJarden1953" class="citation journal cs1">Jarden, Dov (1953). "Curiosa No. 339: A simple proof that a power of an irrational number to an irrational exponent may be rational". <i>Scripta Mathematica</i>. <b>19</b>: 229.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scripta+Mathematica&rft.atitle=Curiosa+No.+339%3A+A+simple+proof+that+a+power+of+an+irrational+number+to+an+irrational+exponent+may+be+rational&rft.volume=19&rft.pages=229&rft.date=1953&rft.aulast=Jarden&rft.aufirst=Dov&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://queuea9.wordpress.com/2015/01/27/the-square-root-of-two-proof/">copy</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorgeVelleman2002" class="citation book cs1">George, Alexander; Velleman, Daniel J. (2002). <a rel="nofollow" class="external text" href="http://condor.depaul.edu/mash/atotheamg.pdf"><i>Philosophies of mathematics</i></a> <span class="cs1-format">(PDF)</span>. Blackwell. pp. 3–4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-631-19544-0" title="Special:BookSources/0-631-19544-0"><bdi>0-631-19544-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophies+of+mathematics&rft.pages=3-4&rft.pub=Blackwell&rft.date=2002&rft.isbn=0-631-19544-0&rft.aulast=George&rft.aufirst=Alexander&rft.au=Velleman%2C+Daniel+J.&rft_id=http%3A%2F%2Fcondor.depaul.edu%2Fmash%2Fatotheamg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Lord, Nick, "Maths bite: irrational powers of irrational numbers can be rational", <i>Mathematical Gazette</i> 92, November 2008, p. 534.</span> </li> <li id="cite_note-Marshall-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-Marshall_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Marshall_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Marshall, Ash J., and Tan, Yiren, "A rational number of the form <i>a</i><sup><i>a</i></sup> with <i>a</i> irrational", <i><a href="/wiki/Mathematical_Gazette" class="mw-redirect" title="Mathematical Gazette">Mathematical Gazette</a></i> 96, March 2012, pp. 106-109.</span> </li> <li id="cite_note-:12-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-:12_32-0">^</a></b></span> <span class="reference-text"><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> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Schanuel%E2%80%99s+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%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaldschmidt2023" class="citation web cs1">Waldschmidt, Michel (2023). <a rel="nofollow" class="external text" href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/OpenPbsNT.pdf">"Some of the most famous open problems in number theory"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Some+of+the+most+famous+open+problems+in+number+theory&rft.date=2023&rft.aulast=Waldschmidt&rft.aufirst=Michel&rft_id=https%3A%2F%2Fwebusers.imj-prg.fr%2F~michel.waldschmidt%2Farticles%2Fpdf%2FOpenPbsNT.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-:4-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-:4_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaldschmidt2022" class="citation web cs1">Waldschmidt, Michel (2022). <a rel="nofollow" class="external text" href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs">"Transcendental Number Theory: recent results and open problems"</a>. <i>Michel Waldschmidt</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Michel+Waldschmidt&rft.atitle=Transcendental+Number+Theory%3A+recent+results+and+open+problems.&rft.date=2022&rft.aulast=Waldschmidt&rft.aufirst=Michel&rft_id=https%3A%2F%2Fwebusers.imj-prg.fr%2F~michel.waldschmidt%2Farticles%2Fpdf%2FTNTOpenPbs&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Bridger2007" class="citation book cs1">Mark Bridger (2007). <i>Real Analysis: A Constructive Approach through Interval Arithmetic</i>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-470-45144-8" title="Special:BookSources/978-1-470-45144-8"><bdi>978-1-470-45144-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+A+Constructive+Approach+through+Interval+Arithmetic&rft.pub=John+Wiley+%26+Sons&rft.date=2007&rft.isbn=978-1-470-45144-8&rft.au=Mark+Bridger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErrett_BishopDouglas_Bridges1985" class="citation book cs1">Errett Bishop; Douglas Bridges (1985). <i>Constructive Analysis</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-15066-8" title="Special:BookSources/0-387-15066-8"><bdi>0-387-15066-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Constructive+Analysis&rft.pub=Springer&rft.date=1985&rft.isbn=0-387-15066-8&rft.au=Errett+Bishop&rft.au=Douglas+Bridges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIrrational+number" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a>, <i>Éléments de Géometrie</i>, Note IV, (1802), Paris</li> <li>Rolf Wallisser, "On Lambert's proof of the irrationality of π", in <i>Algebraic Number Theory and Diophantine Analysis</i>, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyter</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrational_number&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Irrational_numbers" class="extiw" title="commons:Category:Irrational numbers">Irrational numbers</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.dm.uniba.it/~psiche/bas2/node5.html">Zeno's Paradoxes and Incommensurability</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160513063302/http://www.dm.uniba.it/~psiche/bas2/node5.html">Archived</a> 2016-05-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (n.d.). 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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:Irrational_number" title="Template:Irrational number"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Irrational_number" title="Template talk:Irrational number"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Irrational_number" title="Special:EditPage/Template:Irrational number"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Irrational_numbers" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Irrational numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chaitin%27s_constant" title="Chaitin's constant">Chaitin's</a> (<span class="texhtml">Ω</span>)</li> <li><a href="/wiki/Liouville_number" title="Liouville number">Liouville</a></li> <li><a href="/wiki/Prime_constant" title="Prime constant">Prime</a> (<span class="texhtml mvar" style="font-style:italic;">ρ</span>)</li> <li><a href="/wiki/Omega_constant" title="Omega constant">Omega</a></li> <li><a href="/wiki/Cahen%27s_constant" title="Cahen's constant">Cahen</a></li></ul> <ul><li><a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">Logarithm of 2</a></li> <li><a href="/wiki/Dottie_number" title="Dottie number">Dottie</a></li> <li><a href="/wiki/Lemniscate_constant" title="Lemniscate constant">Lemniscate</a> (<span class="texhtml mvar" style="font-style:italic;">ϖ</span>)</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> (<span class="texhtml"><i>ζ</i>(3)</span>)</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> (<span class="texhtml mvar" style="font-style:italic;">ρ</span>)</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> (<span class="texhtml mvar" style="font-style:italic;">ψ</span>)</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Borwein_constant" title="Erdős–Borwein constant">Erdős–Borwein</a> (<span class="texhtml mvar" style="font-style:italic;">E</span>)</li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a> (<span class="texhtml mvar" style="font-style:italic;">φ</span>)</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> (<span class="texhtml mvar" style="font-style:italic;">ς</span>)</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> (<span class="texhtml"><i>δ</i><sub><i>S</i></sub></span>)</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> (<span class="texhtml mvar" style="font-style:italic;">e</span>)</li> <li><a href="/wiki/Pi" title="Pi">Pi</a> (<span class="texhtml mvar" style="font-style:italic;">π</span>)</li></ul> </div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Gold,_square_root_of_2,_and_square_root_of_3_rectangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/50px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png" decoding="async" width="50" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/75px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg/100px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.svg.png 2x" data-file-width="980" data-file-height="1755" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schizophrenic_number" title="Schizophrenic number">Schizophrenic</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a></li> <li><a href="/wiki/Trigonometric_number" class="mw-redirect" title="Trigonometric number">Trigonometric</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Number_systems" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q607728#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" 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