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vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">gøym</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">Byrjinga</div> </a> </li> <li id="toc-Eigenskapar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eigenskapar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Eigenskapar</span> </div> </a> <button aria-controls="toc-Eigenskapar-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>Vis/skjul underseksjonen Eigenskapar</span> </button> <ul id="toc-Eigenskapar-sublist" class="vector-toc-list"> <li id="toc-Det_finst_irrasjonelle_tal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Det_finst_irrasjonelle_tal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Det finst irrasjonelle tal</span> </div> </a> <ul id="toc-Det_finst_irrasjonelle_tal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dei_irrasjonelle_tala_dannar_inga_vanleg_algebraisk_struktur" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dei_irrasjonelle_tala_dannar_inga_vanleg_algebraisk_struktur"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Dei irrasjonelle tala dannar inga vanleg algebraisk struktur</span> </div> </a> <ul id="toc-Dei_irrasjonelle_tala_dannar_inga_vanleg_algebraisk_struktur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dei_irrasjonelle_tala_har_inga_periodisk_desimalutvikling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dei_irrasjonelle_tala_har_inga_periodisk_desimalutvikling"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Dei irrasjonelle tala har inga periodisk desimalutvikling</span> </div> </a> <ul id="toc-Dei_irrasjonelle_tala_har_inga_periodisk_desimalutvikling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sjå_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sjå_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Sjå også</span> </div> </a> <ul id="toc-Sjå_også-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="Innhald" 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="Vis/skjul innhaldslista" > <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">Vis/skjul innhaldslista</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">Irrasjonale tal</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="Gå til ein artikkel på eit anna språk. Tilgjengeleg på 89 språk" > <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 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Irrasjonalt_tall" title="Irrasjonalt tall – norsk bokmål" lang="nb" hreflang="nb" data-title="Irrasjonalt tall" data-language-autonym="Norsk bokmål" data-language-local-name="norsk bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Irrationella_tal" title="Irrationella tal – svensk" lang="sv" hreflang="sv" data-title="Irrationella tal" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</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-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 – enaresamisk" lang="smn" hreflang="smn" data-title="Irrationaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="enaresamisk" 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 – arabisk" lang="ar" hreflang="ar" data-title="عدد غير كسري" data-language-autonym="العربية" data-language-local-name="arabisk" 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 – assamesisk" lang="as" hreflang="as" data-title="অপৰিমেয় সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="assamesisk" 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 – asturisk" lang="ast" hreflang="ast" data-title="Númberu irracional" data-language-autonym="Asturianu" data-language-local-name="asturisk" 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 – aserbajdsjansk" lang="az" hreflang="az" data-title="İrrasional ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" 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 – 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-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 – basjkirsk" lang="ba" hreflang="ba" data-title="Иррациональ һан" data-language-autonym="Башҡортса" data-language-local-name="basjkirsk" 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 – belarusisk" lang="be" hreflang="be" data-title="Ірацыянальны лік" data-language-autonym="Беларуская" data-language-local-name="belarusisk" 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 – bulgarsk" lang="bg" hreflang="bg" data-title="Ирационално число" data-language-autonym="Български" data-language-local-name="bulgarsk" 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 – bosnisk" lang="bs" hreflang="bs" data-title="Iracionalan broj" data-language-autonym="Bosanski" data-language-local-name="bosnisk" 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 – katalansk" lang="ca" hreflang="ca" data-title="Nombre irracional" data-language-autonym="Català" data-language-local-name="katalansk" 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 – tsjuvansk" lang="cv" hreflang="cv" data-title="Иррационаллă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvansk" 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 – tsjekkisk" lang="cs" hreflang="cs" data-title="Iracionální číslo" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" 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 – walisisk" lang="cy" hreflang="cy" data-title="Rhif anghymarebol" data-language-autonym="Cymraeg" data-language-local-name="walisisk" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Irrationale_Zahl" title="Irrationale Zahl – tysk" lang="de" hreflang="de" data-title="Irrationale Zahl" data-language-autonym="Deutsch" data-language-local-name="tysk" 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 – estisk" lang="et" hreflang="et" data-title="Irratsionaalarvud" data-language-autonym="Eesti" data-language-local-name="estisk" 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 – gresk" lang="el" hreflang="el" data-title="Άρρητος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="gresk" 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 – engelsk" lang="en" hreflang="en" data-title="Irrational number" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – spansk" lang="es" hreflang="es" data-title="Número irracional" data-language-autonym="Español" data-language-local-name="spansk" 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 – baskisk" lang="eu" hreflang="eu" data-title="Zenbaki irrazional" data-language-autonym="Euskara" data-language-local-name="baskisk" 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 – persisk" lang="fa" hreflang="fa" data-title="عدد گنگ" data-language-autonym="فارسی" data-language-local-name="persisk" 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 – færøysk" lang="fo" hreflang="fo" data-title="Irrationell tøl" data-language-autonym="Føroyskt" data-language-local-name="færøysk" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="betre artikkel"><a href="https://fr.wikipedia.org/wiki/Nombre_irrationnel" title="Nombre irrationnel – fransk" lang="fr" hreflang="fr" data-title="Nombre irrationnel" data-language-autonym="Français" data-language-local-name="fransk" 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 – irsk" lang="ga" hreflang="ga" data-title="Uimhir éagóimheasta" data-language-autonym="Gaeilge" data-language-local-name="irsk" 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 – galisisk" lang="gl" hreflang="gl" data-title="Número irracional" data-language-autonym="Galego" data-language-local-name="galisisk" 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 – koreansk" lang="ko" hreflang="ko" data-title="무리수" data-language-autonym="한국어" data-language-local-name="koreansk" 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 – armensk" lang="hy" hreflang="hy" data-title="Իռացիոնալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="armensk" 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 – kroatisk" lang="hr" hreflang="hr" data-title="Iracionalni broj" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" 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 – indonesisk" lang="id" hreflang="id" data-title="Bilangan irasional" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" 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 – islandsk" lang="is" hreflang="is" data-title="Óræðar tölur" data-language-autonym="Íslenska" data-language-local-name="islandsk" 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 – italiensk" lang="it" hreflang="it" data-title="Numero irrazionale" data-language-autonym="Italiano" data-language-local-name="italiensk" 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 – hebraisk" lang="he" hreflang="he" data-title="מספר אי-רציונלי" data-language-autonym="עברית" data-language-local-name="hebraisk" 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 – georgisk" lang="ka" hreflang="ka" data-title="ირაციონალური რიცხვი" data-language-autonym="ქართული" data-language-local-name="georgisk" 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 – kasakhisk" lang="kk" hreflang="kk" data-title="Рабайсыз сан" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" 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 – kurdisk" lang="ku" hreflang="ku" data-title="Hejmarên na aqlî" data-language-autonym="Kurdî" data-language-local-name="kurdisk" 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 – kirgisisk" lang="ky" hreflang="ky" data-title="Иррационалдык сан" data-language-autonym="Кыргызча" data-language-local-name="kirgisisk" 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 – laotisk" lang="lo" hreflang="lo" data-title="ຈຳນວນອະປົກກະຕິ" data-language-autonym="ລາວ" data-language-local-name="laotisk" 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 – latvisk" lang="lv" hreflang="lv" data-title="Iracionāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="latvisk" 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 – litauisk" lang="lt" hreflang="lt" data-title="Iracionalusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="litauisk" 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 – lombardisk" lang="lmo" hreflang="lmo" data-title="Nümar irazziunaal" data-language-autonym="Lombard" data-language-local-name="lombardisk" 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 – ungarsk" lang="hu" hreflang="hu" data-title="Irracionális számok" data-language-autonym="Magyar" data-language-local-name="ungarsk" 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 – makedonsk" lang="mk" hreflang="mk" data-title="Ирационален број" data-language-autonym="Македонски" data-language-local-name="makedonsk" 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 – madagassisk" lang="mg" hreflang="mg" data-title="Isa tsivoasaina" data-language-autonym="Malagasy" data-language-local-name="madagassisk" 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 – maltesisk" lang="mt" hreflang="mt" data-title="Numru irrazzjonali" data-language-autonym="Malti" data-language-local-name="maltesisk" 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 – malayisk" lang="ms" hreflang="ms" data-title="Nombor bukan nisbah" data-language-autonym="Bahasa Melayu" data-language-local-name="malayisk" 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 – mongolsk" lang="mn" hreflang="mn" data-title="Иррационал тоо" data-language-autonym="Монгол" data-language-local-name="mongolsk" 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 – nederlandsk" lang="nl" hreflang="nl" data-title="Irrationaal getal" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" 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 – japansk" lang="ja" hreflang="ja" data-title="無理数" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</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 – usbekisk" lang="uz" hreflang="uz" data-title="Irratsional sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbekisk" 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 – panjabi" lang="pa" hreflang="pa" data-title="ਗ਼ੈਰ-ਬਟੇਨੁਮਾ ਸੰਖਿਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" 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 – polsk" lang="pl" hreflang="pl" data-title="Liczby niewymierne" data-language-autonym="Polski" data-language-local-name="polsk" 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 – portugisisk" lang="pt" hreflang="pt" data-title="Número irracional" data-language-autonym="Português" data-language-local-name="portugisisk" 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 – rumensk" lang="ro" hreflang="ro" data-title="Număr irațional" data-language-autonym="Română" data-language-local-name="rumensk" 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 – russisk" lang="ru" hreflang="ru" data-title="Иррациональное число" data-language-autonym="Русский" data-language-local-name="russisk" 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 – albansk" lang="sq" hreflang="sq" data-title="Numrat irracionalë" data-language-autonym="Shqip" data-language-local-name="albansk" 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 – siciliansk" lang="scn" hreflang="scn" data-title="Nùmmuru irrazziunali" data-language-autonym="Sicilianu" data-language-local-name="siciliansk" 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 – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Irrational number" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" 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 – slovakisk" lang="sk" hreflang="sk" data-title="Iracionálne číslo" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" 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 – slovensk" lang="sl" hreflang="sl" data-title="Iracionalno število" data-language-autonym="Slovenščina" data-language-local-name="slovensk" 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 – sorani" lang="ckb" hreflang="ckb" data-title="ژمارەی ناڕێژەیی" data-language-autonym="کوردی" data-language-local-name="sorani" 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 – serbisk" lang="sr" hreflang="sr" data-title="Ирационалан број" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" 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 – serbokroatisk" lang="sh" hreflang="sh" data-title="Iracionalni broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" 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 – finsk" lang="fi" hreflang="fi" data-title="Irrationaaliluku" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</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-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-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 – tadsjikisk" lang="tg" hreflang="tg" data-title="Ададҳои ирратсионалӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadsjikisk" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0rrasyonel_say%C4%B1lar" title="İrrasyonel sayılar – tyrkisk" lang="tr" hreflang="tr" data-title="İrrasyonel sayılar" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" 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 – ukrainsk" lang="uk" hreflang="uk" data-title="Ірраціональне число" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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-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 – vietnamesisk" lang="vi" hreflang="vi" data-title="Số vô tỉ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Irratsionaalarv" title="Irratsionaalarv – sørestisk" lang="vro" hreflang="vro" data-title="Irratsionaalarv" data-language-autonym="Võro" data-language-local-name="sørestisk" class="interlanguage-link-target"><span>Võro</span></a></li><li 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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">Frå Wikipedia – det frie oppslagsverket</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="nn" dir="ltr"><p><b>Irrasjonelle tal</b> er <a href="/wiki/Reelle_tal" title="Reelle tal">reelle tal</a> som <i>ikkje</i> kan skrivast på <a href="/wiki/Br%C3%B8k" title="Brøk">brøkform</a> som <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 {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48d87468620ad6c70385ddd0d024577ccb559e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.877ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}}}"></span>, der <i>m</i> er eit <a href="/wiki/Heiltal" title="Heiltal">heiltal</a> og <i>n</i> eit <a href="/wiki/Naturlege_tal" class="mw-redirect" title="Naturlege tal">naturleg tal</a>. Døme er <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> og <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>. <a href="/wiki/Mengde" class="mw-redirect" title="Mengde">Mengda</a> av irrasjonelle tal vert stundom på symbolform uttrykt som <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} -\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} -\mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39afc7bb32c752afe2ff0225eaec7803ebe04ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.327ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} -\mathbb {Q} }"></span>, men formar ikkje ein vanleg <a href="/wiki/Algebraisk_struktur" title="Algebraisk struktur">algebraisk struktur</a>. Dei rasjonelle tala er ei <a href="/w/index.php?title=Tett_(matematikk)&action=edit&redlink=1" class="new" title="Tett (matematikk) (sida finst ikkje)">tett</a>, <a href="/w/index.php?title=Utellbar&action=edit&redlink=1" class="new" title="Utellbar (sida finst ikkje)">utellbar</a> og ikkje-samanhengande delmengd av <a href="/wiki/Reelle_tal" title="Reelle tal">dei reelle tala</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Eigenskapar">Eigenskapar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrasjonale_tal&veaction=edit&section=1" title="Endre bolk: Eigenskapar" class="mw-editsection-visualeditor"><span>endre</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Irrasjonale_tal&action=edit&section=1" title="Rediger kildekoden til seksjonen Eigenskapar"><span>endre wikiteksten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Det_finst_irrasjonelle_tal">Det finst irrasjonelle tal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrasjonale_tal&veaction=edit&section=2" title="Endre bolk: Det finst irrasjonelle tal" class="mw-editsection-visualeditor"><span>endre</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Irrasjonale_tal&action=edit&section=2" title="Rediger kildekoden til seksjonen Det finst irrasjonelle tal"><span>endre wikiteksten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dersom <i>N</i> ikkje er eit kvadrattal, så er <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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3d8948b2c154ccc7f7523b035ae7e254e04190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.999ex; height:3.009ex;" alt="{\displaystyle {\sqrt {N}}}"></span> eit irrasjonelt tal. Spesielt er <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> irrasjonelt. Eit geometrisk argument for dette vart funne for 2500 år sidan av pytagorearen Hippasus av Metapontum (eller, det er i alle fall han som har vorte tilegna funnet). Eit meir moderne <a href="/w/index.php?title=Prov&action=edit&redlink=1" class="new" title="Prov (sida finst ikkje)">prov</a> er ved <a href="/w/index.php?title=Sj%C3%B8lvmotseiingsbevis&action=edit&redlink=1" class="new" title="Sjølvmotseiingsbevis (sida finst ikkje)">motsegn</a>: Anta at √2 = m/n, der m og n er naturlege tal (me veit at √2 er positiv) og m + n minst mogleg. Då er også <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}}={\frac {2n-m}{m-n}}={\frac {2n-{\sqrt {2}}n}{2n-n}}={\frac {2-{\sqrt {2}}}{{\sqrt {2}}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>n</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}={\frac {2n-m}{m-n}}={\frac {2n-{\sqrt {2}}n}{2n-n}}={\frac {2-{\sqrt {2}}}{{\sqrt {2}}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13b6dec00ffd5e4a7ef15ce48b67720bff24fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.332ex; height:6.843ex;" alt="{\displaystyle {\sqrt {2}}={\frac {2n-m}{m-n}}={\frac {2n-{\sqrt {2}}n}{2n-n}}={\frac {2-{\sqrt {2}}}{{\sqrt {2}}-1}}}"></span>, men <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 2n-m+m-n=n<m+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>m</mi> <mo>−</mo> <mi>n</mi> <mo>=</mo> <mi>n</mi> <mo><</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n-m+m-n=n<m+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c0a0d0af2c710486bfdd431ee6d0efb002a2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.421ex; height:2.343ex;" alt="{\displaystyle 2n-m+m-n=n<m+n}"></span>. </p><p>Anta at det finst naturlege tal <i>m</i> og <i>n</i> slik at <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}}=m/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=m/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b96437672777d9d8a11cfcca9bf80bd135448881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2}}=m/n}"></span>. Me vel <i>m</i> og <i>n</i> slik at <i>n</i> er minst mogleg. Då må <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 2n^{2}=m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n^{2}=m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0aee265b88d4b6a8595d6839c42aa6ad5af810a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.805ex; height:2.676ex;" alt="{\displaystyle 2n^{2}=m^{2}}"></span>, så 2 må dela <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 m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d80831ded84ee5d9e1708e304c8868aa246409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle m^{2}}"></span>. Då må 2 også dela <i>m</i>, så 4 deler <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 m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d80831ded84ee5d9e1708e304c8868aa246409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle m^{2}}"></span> og dermed også <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 2n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0852ba6ce03ff07c4e4a64fb7eb648dd65a32443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.611ex; height:2.676ex;" alt="{\displaystyle 2n^{2}}"></span>. Det fylgjer at 2 også deler <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"></span> og dermed også <i>n</i>, så både <i>m</i> og <i>n</i> er <a href="/wiki/Partal" class="mw-redirect" title="Partal">partal</a>. Dette vil seia at me også har <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}}=(m/2)/(n/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=(m/2)/(n/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318ce00e32a5ab57343da0178a4ee03f939f38f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.063ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2}}=(m/2)/(n/2)}"></span> med naturlege tal <i>m</i>/2 og <i>n</i>/2. Dette er eit motsegn, sidan her er <a href="/w/index.php?title=Teljaren&action=edit&redlink=1" class="new" title="Teljaren (sida finst ikkje)">teljaren</a> endå mindre enn i brøken me starta med. </p><p>Eit anna irrasjonelt tal er <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 log2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle log2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5e590a06f9391989c181589278b1abb2e9aa56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.099ex; height:2.509ex;" alt="{\displaystyle log2}"></span>. Me har igjen eit motseiingsprov: </p><p>Anta at et finst naturlege tal <i>m</i> og <i>n</i> slik at <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}=m/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log {2}=m/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0096a0b44aca8d246778f6d00c8f579a1b7f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.217ex; height:2.843ex;" alt="{\displaystyle \log {2}=m/n}"></span>. Me vel <i>m</i> og <i>n</i> slik at <i>n</i> er minst mogleg. Då må <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^{m/n}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</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>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{m/n}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc222ecd99e0eb0d10050ab841dd1aa41ae1a106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.069ex; height:2.843ex;" alt="{\displaystyle 10^{m/n}=2}"></span>, så <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}\cdot 5^{m}=2^{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>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}\cdot 5^{m}=2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6bd38430dd48b294a384fbfb186f51a3d125e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.833ex; height:2.343ex;" alt="{\displaystyle 2^{m}\cdot 5^{m}=2^{n}}"></span>. Sidan 2 er eit <a href="/wiki/Primtal" title="Primtal">primtal</a>, så må då <i>m</i> = 0 og dermed log 2 = 0, noko som ikkje er tilfelle. </p><p>Det skal seiast at begge desse prova fyrst vert fullstendige når me har vist at dei gjevne tala faktisk eksisterer. For kvadratrota av 2 finst eit enkelt, geometrisk argument: <a href="/wiki/Hypotenus" title="Hypotenus">Hypotenusen</a> x til ein rettvinkla <a href="/wiki/Trekant" title="Trekant">trekant</a> med <a href="/wiki/Katet" title="Katet">katetar</a> 1 og 1 tilfredsstiller 2 = x<sup>2</sup>. Sidan hypotenusen eksisterer, så eksisterer dermed også ei kvadratrot til 2. Eit argument som byggjer direkte på <a href="/wiki/Komplettleiksprinsippet" title="Komplettleiksprinsippet">komplettleiksprinsippet</a> finst også: Studerer me mengda <i>A</i> = {<i>x</i> : <i>x</i> <sup>2</sup> < 2}, <i>x</i> reelle tal, så må denne ha ei <a href="/w/index.php?title=Minste_%C3%B8vre_grense&action=edit&redlink=1" class="new" title="Minste øvre grense (sida finst ikkje)">minste øvre grense</a> sup <i>A</i>. Det er mogleg å visa at sup <i>A</i> korkje er med i <i>A</i> eller <i>A</i> = {x : x <sup>2</sup> > 2 og x > 0}, så difor må (sup A)^2 = 2. (Dersom sup A er med i A, så finst det eit tal høgare enn sup A som også er med i A', og dersom sup A er med i A', så finst det eit tal lågare enn sup A som også er med i A'. Dette strid mot at sup A er den minste øvre grensa til A.) </p> <div class="mw-heading mw-heading3"><h3 id="Dei_irrasjonelle_tala_dannar_inga_vanleg_algebraisk_struktur">Dei irrasjonelle tala dannar inga vanleg algebraisk struktur</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrasjonale_tal&veaction=edit&section=3" title="Endre bolk: Dei irrasjonelle tala dannar inga vanleg algebraisk struktur" class="mw-editsection-visualeditor"><span>endre</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Irrasjonale_tal&action=edit&section=3" title="Rediger kildekoden til seksjonen Dei irrasjonelle tala dannar inga vanleg algebraisk struktur"><span>endre wikiteksten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dei irrasjonelle tala er korkje <a href="/w/index.php?title=Lukka_(matematikk)&action=edit&redlink=1" class="new" title="Lukka (matematikk) (sida finst ikkje)">lukka</a> under <a href="/wiki/Addisjon" title="Addisjon">addisjon</a> eller <a href="/wiki/Multiplikasjon" title="Multiplikasjon">multiplikasjon</a>: </p> <ul><li><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+{\sqrt {2}})+(2-{\sqrt {2}})=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+{\sqrt {2}})+(2-{\sqrt {2}})=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2fa97f9eb6a2e54bfb12c7a7f777ea6ec5d972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.922ex; height:3.176ex;" alt="{\displaystyle (2+{\sqrt {2}})+(2-{\sqrt {2}})=4}"></span> er ikkje eit irrasjonelt tal, sjølv om begge <a href="/w/index.php?title=Addend&action=edit&redlink=1" class="new" title="Addend (sida finst ikkje)">addendane</a> er det.</li> <li><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+{\sqrt {2}})\cdot (2-{\sqrt {2}})=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+{\sqrt {2}})\cdot (2-{\sqrt {2}})=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e73c43feebaebded670693ffa13ba140053c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.761ex; height:3.176ex;" alt="{\displaystyle (2+{\sqrt {2}})\cdot (2-{\sqrt {2}})=2}"></span> er ikkje eit irrasjonelt tal, sjølv om begge <a href="/w/index.php?title=Faktor&action=edit&redlink=1" class="new" title="Faktor (sida finst ikkje)">faktorane</a> er det.</li></ul> <p>I tillegg kan me observera at det heller ikkje er tilfelle at dersom <i>a</i> og <i>b</i> er irrasjonelle, så er nødvendigvis <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^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921151d29231ebd65eea7632a88215273a32234c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.167ex; height:2.676ex;" alt="{\displaystyle a^{b}}"></span> irrasjonell. Me ser på <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span>: </p> <ul><li>Dersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span> er rasjonell, så er resultatet vist umiddelbart.</li> <li>Dersom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span> er irrasjonell, så er <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04343bbf8200d6b70c7c0fd80afb106219af2226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.015ex; height:3.843ex;" alt="{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}=2}"></span> det ettersøkte moteksemplet.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Dei_irrasjonelle_tala_har_inga_periodisk_desimalutvikling">Dei irrasjonelle tala har inga periodisk desimalutvikling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrasjonale_tal&veaction=edit&section=4" title="Endre bolk: Dei irrasjonelle tala har inga periodisk desimalutvikling" class="mw-editsection-visualeditor"><span>endre</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Irrasjonale_tal&action=edit&section=4" title="Rediger kildekoden til seksjonen Dei irrasjonelle tala har inga periodisk desimalutvikling"><span>endre wikiteksten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I motsetnad til <a href="/wiki/Rasjonelle_tal" class="mw-redirect" title="Rasjonelle tal">rasjonelle tal</a> har ikkje irrasjonelle tal ei periodisk desimalutvikling. Beviset går ut på å gå ut frå at ei slik desimalutvikling finst og visa at då må talet vera rasjonelt. </p> <div class="mw-heading mw-heading2"><h2 id="Sjå_også"><span id="Sj.C3.A5_ogs.C3.A5"></span>Sjå også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Irrasjonale_tal&veaction=edit&section=5" title="Endre bolk: Sjå også" class="mw-editsection-visualeditor"><span>endre</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Irrasjonale_tal&action=edit&section=5" title="Rediger kildekoden til seksjonen Sjå også"><span>endre wikiteksten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Inkommensurabilitet" class="mw-redirect" title="Inkommensurabilitet">Inkommensurabilitet</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r3515230">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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