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meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Masuk_log&returnto=Bilangan+irasional" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Opsi lainnya" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas pribadi" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Bilangan+irasional" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Bilangan+irasional" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Sejarah</span> </div> </a> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-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="Daftar isi" 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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Bilangan irasional</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="Pergi ke artikel dalam bahasa lain. Terdapat 89 bahasa" > <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 bahasa</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-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 – Arab" lang="ar" hreflang="ar" data-title="عدد غير كسري" data-language-autonym="العربية" data-language-local-name="Arab" 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 – Assam" lang="as" hreflang="as" data-title="অপৰিমেয় সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assam" 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 – Asturia" lang="ast" hreflang="ast" data-title="Númberu irracional" data-language-autonym="Asturianu" data-language-local-name="Asturia" 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-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 – Belarusia" lang="be" hreflang="be" data-title="Ірацыянальны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusia" 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 – Bulgaria" lang="bg" hreflang="bg" data-title="Ирационално число" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</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 – Bengali" lang="bn" hreflang="bn" data-title="অমূলদ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Iracionalan_broj" title="Iracionalan broj – Bosnia" lang="bs" hreflang="bs" data-title="Iracionalan broj" data-language-autonym="Bosanski" data-language-local-name="Bosnia" 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 – Katalan" lang="ca" hreflang="ca" data-title="Nombre irracional" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</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 – Kurdi Sorani" lang="ckb" hreflang="ckb" data-title="ژمارەی ناڕێژەیی" data-language-autonym="کوردی" data-language-local-name="Kurdi Sorani" class="interlanguage-link-target"><span>کوردی</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 – Cheska" lang="cs" hreflang="cs" data-title="Iracionální číslo" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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-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 – Dansk" lang="da" hreflang="da" data-title="Irrationale tal" data-language-autonym="Dansk" data-language-local-name="Dansk" 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 – Jerman" lang="de" hreflang="de" data-title="Irrationale Zahl" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</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 – Yunani" lang="el" hreflang="el" data-title="Άρρητος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Irrational_number" title="Irrational number – Inggris" lang="en" hreflang="en" data-title="Irrational number" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – Spanyol" lang="es" hreflang="es" data-title="Número irracional" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Irratsionaalarvud" title="Irratsionaalarvud – Esti" lang="et" hreflang="et" data-title="Irratsionaalarvud" data-language-autonym="Eesti" data-language-local-name="Esti" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/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 – Persia" lang="fa" hreflang="fa" data-title="عدد گنگ" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Irrationaaliluku" title="Irrationaaliluku – Suomi" lang="fi" hreflang="fi" data-title="Irrationaaliluku" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Irratsionaalarv" title="Irratsionaalarv – Võro" lang="vro" hreflang="vro" data-title="Irratsionaalarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</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 – Faroe" lang="fo" hreflang="fo" data-title="Irrationell tøl" data-language-autonym="Føroyskt" data-language-local-name="Faroe" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="artikel bagus"><a href="https://fr.wikipedia.org/wiki/Nombre_irrationnel" title="Nombre irrationnel – Prancis" lang="fr" hreflang="fr" data-title="Nombre irrationnel" data-language-autonym="Français" data-language-local-name="Prancis" 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 – Irlandia" lang="ga" hreflang="ga" data-title="Uimhir éagóimheasta" data-language-autonym="Gaeilge" data-language-local-name="Irlandia" 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 – Galisia" lang="gl" hreflang="gl" data-title="Número irracional" data-language-autonym="Galego" data-language-local-name="Galisia" class="interlanguage-link-target"><span>Galego</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 – Ibrani" lang="he" hreflang="he" data-title="מספר אי-רציונלי" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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 – Kroasia" lang="hr" hreflang="hr" data-title="Iracionalni broj" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</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 – Hungaria" lang="hu" hreflang="hu" data-title="Irracionális számok" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%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 – Armenia" lang="hy" hreflang="hy" data-title="Իռացիոնալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenia" class="interlanguage-link-target"><span>Հայերեն</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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%93r%C3%A6%C3%B0ar_t%C3%B6lur" title="Óræðar tölur – Islandia" lang="is" hreflang="is" data-title="Óræðar tölur" data-language-autonym="Íslenska" data-language-local-name="Islandia" 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 – Italia" lang="it" hreflang="it" data-title="Numero irrazionale" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%84%A1%E7%90%86%E6%95%B0" title="無理数 – Jepang" lang="ja" hreflang="ja" data-title="無理数" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</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 – Georgia" lang="ka" hreflang="ka" data-title="ირაციონალური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수 – Korea" lang="ko" hreflang="ko" data-title="무리수" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</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 – Kurdi" lang="ku" hreflang="ku" data-title="Hejmarên na aqlî" data-language-autonym="Kurdî" data-language-local-name="Kurdi" 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 – Kirgiz" lang="ky" hreflang="ky" data-title="Иррационалдык сан" data-language-autonym="Кыргызча" data-language-local-name="Kirgiz" class="interlanguage-link-target"><span>Кыргызча</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-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-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-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Iracionalusis_skai%C4%8Dius" title="Iracionalusis skaičius – Lituavi" lang="lt" hreflang="lt" data-title="Iracionalusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lituavi" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Iracion%C4%81ls_skaitlis" title="Iracionāls skaitlis – Latvi" lang="lv" hreflang="lv" data-title="Iracionāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvi" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsivoasaina" title="Isa tsivoasaina – Malagasi" lang="mg" hreflang="mg" data-title="Isa tsivoasaina" data-language-autonym="Malagasy" data-language-local-name="Malagasi" class="interlanguage-link-target"><span>Malagasy</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 – Makedonia" lang="mk" hreflang="mk" data-title="Ирационален број" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%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-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 – Mongolia" lang="mn" hreflang="mn" data-title="Иррационал тоо" data-language-autonym="Монгол" data-language-local-name="Mongolia" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%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 – Melayu" lang="ms" hreflang="ms" data-title="Nombor bukan nisbah" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Numru_irrazzjonali" title="Numru irrazzjonali – Malta" lang="mt" hreflang="mt" data-title="Numru irrazzjonali" data-language-autonym="Malti" data-language-local-name="Malta" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Irrationaal_getal" title="Irrationaal getal – Belanda" lang="nl" hreflang="nl" data-title="Irrationaal getal" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Irrasjonale_tal" title="Irrasjonale tal – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Irrasjonale tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Irrasjonalt_tall" title="Irrasjonalt tall – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Irrasjonalt tall" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%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 – Oriya" lang="or" hreflang="or" data-title="ଅପରିମେୟ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Oriya" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</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 – Polski" lang="pl" hreflang="pl" data-title="Liczby niewymierne" data-language-autonym="Polski" data-language-local-name="Polski" 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 – Portugis" lang="pt" hreflang="pt" data-title="Número irracional" data-language-autonym="Português" data-language-local-name="Portugis" 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 – Rumania" lang="ro" hreflang="ro" data-title="Număr irațional" data-language-autonym="Română" data-language-local-name="Rumania" 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 – Rusia" lang="ru" hreflang="ru" data-title="Иррациональное число" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</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 – Sisilia" lang="scn" hreflang="scn" data-title="Nùmmuru irrazziunali" data-language-autonym="Sicilianu" data-language-local-name="Sisilia" class="interlanguage-link-target"><span>Sicilianu</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-Kroasia" lang="sh" hreflang="sh" data-title="Iracionalni broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – Sloven" lang="sl" hreflang="sl" data-title="Iracionalno število" data-language-autonym="Slovenščina" data-language-local-name="Sloven" class="interlanguage-link-target"><span>Slovenščina</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-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_irracional%C3%AB" title="Numrat irracionalë – Albania" lang="sq" hreflang="sq" data-title="Numrat irracionalë" data-language-autonym="Shqip" data-language-local-name="Albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%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 – Serbia" lang="sr" hreflang="sr" data-title="Ирационалан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Irrationella_tal" title="Irrationella tal – Swedia" lang="sv" hreflang="sv" data-title="Irrationella tal" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/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-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BF%E0%AE%95%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="விகிதமுறா எண் – Tamil" lang="ta" hreflang="ta" data-title="விகிதமுறா எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%A8%E0%B0%BF%E0%B0%B7%E0%B1%8D%E0%B0%AA_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="అనిష్ప సంఖ్య – Telugu" lang="te" hreflang="te" data-title="అనిష్ప సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D2%B3%D0%BE%D0%B8_%D0%B8%D1%80%D1%80%D0%B0%D1%82%D1%81%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D3%A3" title="Ададҳои ирратсионалӣ – Tajik" lang="tg" hreflang="tg" data-title="Ададҳои ирратсионалӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AD%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A2%E0%B8%B0" title="จำนวนอตรรกยะ – Thai" lang="th" hreflang="th" data-title="จำนวนอตรรกยะ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0rrasyonel_say%C4%B1lar" title="İrrasyonel sayılar – Turki" lang="tr" hreflang="tr" data-title="İrrasyonel sayılar" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Ірраціональне число – Ukraina" lang="uk" hreflang="uk" data-title="Ірраціональне число" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%BA%DB%8C%D8%B1%D9%86%D8%A7%D8%B7%D9%82_%D8%B9%D8%AF%D8%AF" title="غیرناطق عدد – Urdu" lang="ur" hreflang="ur" data-title="غیرناطق عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_v%C3%B4_t%E1%BB%89" title="Số vô tỉ – Vietnam" lang="vi" hreflang="vi" data-title="Số vô tỉ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Irrationoale_getalln" title="Irrationoale getalln – West Flemish" lang="vls" hreflang="vls" data-title="Irrationoale getalln" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_al%C3%A1%C3%ACn%C3%AD%C3%ACp%C3%ADn" title="Nọ́mbà aláìníìpín – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà aláìníìpín" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數 – Tionghoa" lang="zh" hreflang="zh" data-title="無理數" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數 – Kanton" lang="yue" hreflang="yue" data-title="無理數" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q607728#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bilangan_irasional" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Pembicaraan:Bilangan_irasional&action=edit&redlink=1" rel="discussion" class="new" title="Pembicaraan halaman isi (halaman belum tersedia) [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bilangan_irasional"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Perkakas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Bilangan_irasional"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Bilangan_irasional" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Bilangan_irasional" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa" title="Daftar semua halaman istimewa [q]" accesskey="q"><span>Halaman istimewa</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&oldid=26217939" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Bilangan_irasional&id=26217939&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBilangan_irasional"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBilangan_irasional"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Bilangan+irasional"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Bilangan_irasional&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bilangan_irasional&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Irrational_numbers" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q607728" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</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="id" dir="ltr"><p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, <b>bilangan irasional</b> adalah <a href="/wiki/Bilangan_riil" title="Bilangan riil">bilangan riil</a> yang tidak bisa dibagi (hasil baginya tidak pernah berhenti). Dalam hal ini, bilangan irasional tidak bisa dinyatakan sebagai <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 {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"></span>, dengan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> sebagai <a href="/wiki/Bilangan_bulat" title="Bilangan bulat">bilangan bulat</a> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> tidak sama dengan nol. Jadi bilangan irasional bukan merupakan <a href="/wiki/Bilangan_rasional" title="Bilangan rasional">bilangan rasional</a>. Contoh yang paling populer dari bilangan irasional adalah bilangan <a href="/wiki/Pi" title="Pi">π</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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>, dan bilangan <a href="/wiki/E_(konstanta_matematika)" title="E (konstanta matematika)">e</a>. </p><p>Bilangan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> sebetulnya tidak tepat, yaitu kurang lebih 3.14, tetapi </p> <dl><dd>= 3,1415926535.... atau</dd> <dd>= 3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510...</dd></dl> <p>Untuk bilangan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>: </p> <dl><dd>= 1,4142135623730950488016887242096.... atau</dd> <dd>= 1,41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73798..</dd></dl> <p>dan untuk bilangan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>: </p> <dl><dd>= 2,7182818....</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sejarah">Sejarah</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_irasional&veaction=edit&section=1" title="Sunting bagian: Sejarah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_irasional&action=edit&section=1" title="Sunting kode sumber bagian: Sejarah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas: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/150px-Square_root_of_2_triangle.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/225px-Square_root_of_2_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Square_root_of_2_triangle.svg/300px-Square_root_of_2_triangle.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Bilangan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6ac02637a190523aa10dde1b52ae41964dfff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.191ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\sqrt {2}}}"></span> adalah bilangan irasional.</figcaption></figure> <p>Dalam doctorate in Absentia-nya pada tahun 1799, <i>A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree </i>(Sebuah bukti baru teorema bahwa setiap fungsi aljabar rasional yang tidak terpisahkan dari satu variabel dapat diselesaikan menjadi faktor nyata pada derajat pertama atau kedua), Gauss memberikan bukti teorema fundamental <a href="/wiki/Aljabar" title="Aljabar">aljabar</a> yang menyatakan bahwa setiap-tiap dari polinomial variabel tunggal bukan-konstanta dengan koefisien kompleks memiliki paling sedikit atau setidaknya satu akar kompleks. Namun banyak matematikawan termasuk <a href="https://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert" class="extiw" title="en:Jean le Rond d'Alembert">Jean le Rond d'Alembert</a> yang memberikan bukti yang salah pada awalnya,dan disertasi Gauss juga banyak mengkritik kerja d'Alembert. </p><p>Namun sekali lagi, ironisnya, dengan menggunakan standar sekarang percobaan milik Gauss tidak dapat diterima, yang menyebabkan penggunaan secara implisit teorema Kurva Jordan di dalam kurva <a href="/wiki/Fraktal" title="Fraktal">fraktal</a>. Bagaimanapun, dia secara berkelanjutan memberikan tiga bukti yang lain,yang terakhir pada 1849 yang dikenal sukar. Upayanya dalam mengklarifikasi konsep mengenai <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a> memang banyak dibicarakan (dari contoh bilangan irasional paling terkenal:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256fc4aeac40bb57320642c5564e5735199dd526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.24ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}"></span>, memecahnya dengan menempatkan minus pada satu tingkat di bawah sumbu imajiner dan x pada sumbu positif real, Gauss mengubah bilangan irasional yang sebelumnya dianggap bilangan <i>antara ada dan tiada</i> menjadi dapat diperhitungkan, lihat secara khusus polar kompleks). </p><p>Gauss juga memberikan kontribusi sangat penting bagi <a href="/wiki/Teori_bilangan" title="Teori bilangan">teori bilangan</a>. Di dalam bukunya pada tahun 1801, <i>Disquisitiones Arithmeticae</i> (<a href="/wiki/Bahasa_Latin" title="Bahasa Latin">bahasa Latin</a>:, Investigasi Aritmetika), yang mana, dalam banyak hal, Gauss memperkenalkan penggunaan notasi ≡ untuk kekongruenan dan menggunakannya dalam presentasi yang baik di dalam <a href="/wiki/Aritmetika" title="Aritmetika">aritmetika</a> modular. </p><p>Abad ke-19 menyaksikan perkembangan cepat konsep <a href="/wiki/Bilangan_imajiner" title="Bilangan imajiner">bilangan imajiner</a> di tangan Abraham de Moivre,dan secara khusus <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, yang menjadikannya lebih berdaya guna. Penyelesaian teori mengenai <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a> pada abad ke-19 membedakan bilangan irasional menjadi <a href="/wiki/Bilangan_aljabar" title="Bilangan aljabar">bilangan aljabar</a> dan transenden. Bukti keberadaan bilangan transenden, dan menjamurnya studi-studi saintifik mengenai teori bilangan irasional telah lama dipikirkan sejak <a href="/wiki/Euclid" class="mw-redirect" title="Euclid">Euclid</a>. Tahun 1872 menyaksikan publikasi dari teori-teori dari <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> (oleh muridnya, Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), dan Richard Dedekind. Meray memulai pada 1869, sama dengan Heine, tetapi teorinya dikutip secara umum pada 1872. </p><p>Pecahan kontinu, yang berhubungan dekat dengan bilangan irasional, mendapat perhatian di tangan Euler, dan akhirnya,fajar abad ke-19 benar-benar dibawa menuju keagungan lewat tulisan-tulisan <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>. Dirichlet juga menambahkan dalam teori umumnya, seperti juga banyak sekali kontributor untuk penerapan mengenai subyek ini. </p> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_irasional&veaction=edit&section=2" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_irasional&action=edit&section=2" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r22254620">.mw-parser-output .portalbox{padding:0}.mw-parser-output .portalborder{border:solid #aaa 1px}.mw-parser-output .portalbox.tleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalbox.tright{margin:0.5em 0 0.5em 1em}.mw-parser-output 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autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r18590415">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox 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terhitung</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/Bilangan_asli" title="Bilangan asli">Bilangan asli</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 \scriptstyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40eac26c488d3257e3fbe63619729673145d228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Bilangan_bulat" title="Bilangan bulat">Bilangan bulat</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 \scriptstyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c672518c0350ca035befd41c26633a2d399431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.096ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Bilangan_rasional" title="Bilangan rasional">Bilangan rasional</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 \scriptstyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaa5ab94a056a5a25944ddf0c52c92a404715ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Bilangan_aljabar" title="Bilangan aljabar">Bilangan aljabar</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 \scriptstyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93d69135c22d5b1b10f65fa49c7ece56ae561fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {\overline {\mathbb {Q} }}}"></span>)</li> <li><a href="/w/index.php?title=Perioda&action=edit&redlink=1" class="new" title="Perioda (halaman belum tersedia)">Perioda</a></li> <li><a href="/w/index.php?title=Bilangan_terkomputasi&action=edit&redlink=1" class="new" title="Bilangan terkomputasi (halaman belum tersedia)">Bilangan terkomputasi</a></li> <li><a href="/w/index.php?title=Bilangan_aritmetis&action=edit&redlink=1" class="new" title="Bilangan aritmetis (halaman belum tersedia)">Bilangan aritmetis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Bilangan riil dan<br />cabangan</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/Bilangan_riil" title="Bilangan riil">Bilangan riil</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 \scriptstyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7df6838b44979c6531f6a0306206fbdb0477ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">Bilangan kompleks</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 \scriptstyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe3a54bb4e56c039e18c3af24ba70ab377f7a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Kuaternion" title="Kuaternion">Kuaternion</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 \scriptstyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d00daea5df233d805f1ec5d5ae84845bac2ad06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Oktonion" title="Oktonion">Oktonion</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 \scriptstyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5cf3960cf7ba384648447c15581d5d4589a6d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {O} }"></span>)</li> <li><a href="/w/index.php?title=Sedenion&action=edit&redlink=1" class="new" title="Sedenion (halaman belum tersedia)">Sedenion</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 \scriptstyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef48a593f4503abeab608e8781ba478b7d1b304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.914ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {S} }"></span>)</li> <li><a href="/w/index.php?title=Aljabar_Cayley%E2%80%93Dickson&action=edit&redlink=1" class="new" title="Aljabar Cayley–Dickson (halaman belum tersedia)">Aljabar Cayley–Dickson</a></li> <li><a href="/w/index.php?title=Bilangan_rangkap&action=edit&redlink=1" class="new" title="Bilangan rangkap (halaman belum tersedia)">Bilangan rangkap</a></li> <li><a href="/w/index.php?title=Bilangan_kompleks_hiperbolik&action=edit&redlink=1" class="new" title="Bilangan kompleks hiperbolik (halaman belum tersedia)">Bilangan kompleks hiperbolik</a></li> <li><a href="/wiki/Bilangan_hiperkompleks" title="Bilangan hiperkompleks">Bilangan hiperkompleks</a></li> <li><a href="/w/index.php?title=Bilangan_superreal&action=edit&redlink=1" class="new" title="Bilangan superreal (halaman belum tersedia)">Bilangan superreal</a></li> <li><a class="mw-selflink selflink">Bilangan irasional</a></li> <li><a href="/wiki/Bilangan_transenden" title="Bilangan transenden">Bilangan transenden</a></li> <li><a href="/w/index.php?title=Bilangan_hiperreal&action=edit&redlink=1" class="new" title="Bilangan hiperreal (halaman belum tersedia)">Bilangan hiperreal</a></li> <li><a href="/w/index.php?title=Bilangan_surreal&action=edit&redlink=1" class="new" title="Bilangan surreal (halaman belum tersedia)">Bilangan surreal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Sistem lain</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/Bilangan_kardinal" title="Bilangan kardinal">Bilangan kardinal</a></li> <li><a href="/wiki/Bilangan_ordinal" title="Bilangan ordinal">Bilangan ordinal</a></li> <li><a href="/w/index.php?title=Bilangan_p-adik&action=edit&redlink=1" class="new" title="Bilangan p-adik (halaman belum tersedia)">Bilangan p-adik</a></li> <li><a href="/w/index.php?title=Bilangan_supernatural&action=edit&redlink=1" class="new" title="Bilangan supernatural (halaman belum tersedia)">Bilangan supernatural</a></li></ul> </div></td></tr></tbody></table></div> 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