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href="#Irreducible_fraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Irreducible fraction</span> </div> </a> <ul id="toc-Irreducible_fraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Embedding_of_integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Embedding_of_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Embedding of integers</span> </div> </a> <ul id="toc-Embedding_of_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Equality</span> </div> </a> <ul id="toc-Equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Ordering</span> </div> </a> <ul id="toc-Ordering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Addition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Addition</span> </div> </a> <ul id="toc-Addition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subtraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Subtraction</span> </div> </a> <ul id="toc-Subtraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplication"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Multiplication</span> </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Inverse</span> </div> </a> <ul id="toc-Inverse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Division</span> </div> </a> <ul id="toc-Division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponentiation_to_integer_power" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponentiation_to_integer_power"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.10</span> <span>Exponentiation to integer power</span> </div> </a> <ul id="toc-Exponentiation_to_integer_power-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Continued_fraction_representation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Continued_fraction_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Continued fraction representation</span> </div> </a> <ul id="toc-Continued_fraction_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other representations</span> </div> </a> <ul id="toc-Other_representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_construction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Formal construction</span> </div> </a> <ul id="toc-Formal_construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Countability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Countability"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Countability</span> </div> </a> <ul id="toc-Countability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Real_numbers_and_topological_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_numbers_and_topological_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Real numbers and topological properties</span> </div> </a> <ul id="toc-Real_numbers_and_topological_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p-adic_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#p-adic_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span><i>p</i>-adic numbers</span> </div> </a> <ul id="toc-p-adic_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rational number</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 119 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-119" 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">119 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Rasionale_getal" title="Rasionale getal – Afrikaans" lang="af" hreflang="af" data-title="Rasionale getal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Rationale_Zahl" title="Rationale Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Rationale Zahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%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="परिमेय संख्या – Angika" lang="anp" hreflang="anp" data-title="परिमेय संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</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_%D9%83%D8%B3%D8%B1%D9%8A" title="عدد كسري – Arabic" lang="ar" hreflang="ar" data-title="عدد كسري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_racional" title="Numero racional – Aragonese" lang="an" hreflang="an" data-title="Numero racional" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%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="পৰিমেয় সংখ্যা – Assamese" lang="as" hreflang="as" data-title="পৰিমেয় সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_racional" title="Númberu racional – Asturian" lang="ast" hreflang="ast" data-title="Númberu racional" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Rasional_%C9%99d%C9%99dl%C9%99r" title="Rasional ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Rasional ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B1%D8%A7%D8%B3%DB%8C%D9%88%D9%86%D8%A7%D9%84_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="راسیونال ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="راسیونال ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%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="মূলদ সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="মূলদ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/P%C3%AD-s%C3%B2%CD%98" title="Pí-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Pí-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%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C_%D2%BB%D0%B0%D0%BD" title="Рациональ һан – Bashkir" lang="ba" hreflang="ba" data-title="Рациональ 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%A0%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="Рацыянальны лік – Belarusian" lang="be" hreflang="be" data-title="Рацыянальны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%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="Рацыянальны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Рацыянальны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%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="Рационално число – Bulgarian" lang="bg" hreflang="bg" data-title="Рационално число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Racionalni_broj" title="Racionalni broj – Bosnian" lang="bs" hreflang="bs" data-title="Racionalni broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_feurek" title="Niver feurek – Breton" lang="br" hreflang="br" data-title="Niver feurek" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_racional" title="Nombre racional – Catalan" lang="ca" hreflang="ca" data-title="Nombre racional" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%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="Рационаллă хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Рационаллă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Racion%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Racionální číslo – Czech" lang="cs" hreflang="cs" data-title="Racionální číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhamba_svinu" title="Nhamba svinu – Shona" lang="sn" hreflang="sn" data-title="Nhamba svinu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cymarebol" title="Rhif cymarebol – Welsh" lang="cy" hreflang="cy" data-title="Rhif cymarebol" 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/Rationale_tal" title="Rationale tal – Danish" lang="da" hreflang="da" data-title="Rationale tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rationale_Zahl" title="Rationale Zahl – German" lang="de" hreflang="de" data-title="Rationale Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ratsionaalarv" title="Ratsionaalarv – Estonian" lang="et" hreflang="et" data-title="Ratsionaalarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A1%CE%B7%CF%84%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Ρητός αριθμός – Greek" lang="el" hreflang="el" data-title="Ρητός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_raziun%C3%A8l" title="Nómmer raziunèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer raziunèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_racional" title="Número racional – Spanish" lang="es" hreflang="es" data-title="Número racional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Racionala_nombro" title="Racionala nombro – Esperanto" lang="eo" hreflang="eo" data-title="Racionala 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_arrazional" title="Zenbaki arrazional – Basque" lang="eu" hreflang="eu" data-title="Zenbaki arrazional" 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%88%DB%8C%D8%A7" title="عدد گویا – Persian" lang="fa" hreflang="fa" data-title="عدد گویا" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Rationelt_tal" title="Rationelt tal – Faroese" lang="fo" hreflang="fo" data-title="Rationelt tal" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_rationnel" title="Nombre rationnel – French" lang="fr" hreflang="fr" data-title="Nombre rationnel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_ch%C3%B3imheasta" title="Uimhir chóimheasta – Irish" lang="ga" hreflang="ga" data-title="Uimhir chóimheasta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_racional" title="Número racional – Galician" lang="gl" hreflang="gl" data-title="Número racional" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E6%95%B8" title="有理數 – Gan" lang="gan" hreflang="gan" data-title="有理數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB_%D1%82%D0%BE%D0%B9%D0%B3" title="Рационал тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Рационал тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수 – Korean" lang="ko" hreflang="ko" data-title="유리수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%A1%D6%81%D5%AB%D5%B8%D5%B6%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Ռացիոնալ թիվ – Armenian" lang="hy" hreflang="hy" data-title="Ռացիոնալ թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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="परिमेय संख्या – Hindi" lang="hi" hreflang="hi" data-title="परिमेय 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/Racionalni_broj" title="Racionalni broj – Croatian" lang="hr" hreflang="hr" data-title="Racionalni broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Racionala_nombro" title="Racionala nombro – Ido" lang="io" hreflang="io" data-title="Racionala 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_rasional" title="Bilangan rasional – Indonesian" lang="id" hreflang="id" data-title="Bilangan rasional" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_rational" title="Numero rational – Interlingua" lang="ia" hreflang="ia" data-title="Numero rational" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Amanani_a-rational" title="Amanani a-rational – Xhosa" lang="xh" hreflang="xh" data-title="Amanani a-rational" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%A6%C3%B0ar_t%C3%B6lur" title="Ræðar tölur – Icelandic" lang="is" hreflang="is" data-title="Ræðar tölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_razionale" title="Numero razionale – Italian" lang="it" hreflang="it" data-title="Numero razionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A8%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99" title="מספר רציונלי – Hebrew" lang="he" hreflang="he" data-title="מספר רציונלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%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="რაციონალური რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="რაციონალური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB_%D1%81%D0%B0%D0%BD" title="Рационал сан – Kazakh" lang="kk" hreflang="kk" data-title="Рационал сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_rasyon%C3%A8l" title="Nonm rasyonèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm rasyonèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</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%9B%E0%BA%BB%E0%BA%81%E0%BA%81%E0%BA%B0%E0%BA%95%E0%BA%B4" title="ຈຳນວນປົກກະຕິ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນປົກກະຕິ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_rationalis" title="Numerus rationalis – Latin" lang="la" hreflang="la" data-title="Numerus rationalis" 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/Racion%C4%81ls_skaitlis" title="Racionāls skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Racionāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Racionalusis_skai%C4%8Dius" title="Racionalusis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Racionalusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Rationaol_getal" title="Rationaol getal – Limburgish" lang="li" hreflang="li" data-title="Rationaol getal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_razonal" title="Numero razonal – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero razonal" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/dilcyna%27u" title="dilcyna'u – Lojban" lang="jbo" hreflang="jbo" data-title="dilcyna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Namba_ez%27Omugerageranyo(Rational_NUmbers)" title="Namba ez'Omugerageranyo(Rational NUmbers) – Ganda" lang="lg" hreflang="lg" data-title="Namba ez'Omugerageranyo(Rational NUmbers)" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_razziunaal" title="Nümar razziunaal – Lombard" lang="lmo" hreflang="lmo" data-title="Nümar razziunaal" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Racion%C3%A1lis_sz%C3%A1mok" title="Racionális számok – Hungarian" lang="hu" hreflang="hu" data-title="Racionális számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%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="Рационален број – Macedonian" lang="mk" hreflang="mk" data-title="Рационален број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_voasaina" title="Isa voasaina – Malagasy" lang="mg" hreflang="mg" data-title="Isa voasaina" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AD%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%A8%E0%B4%95%E0%B4%82" title="ഭിന്നകം – Malayalam" lang="ml" hreflang="ml" data-title="ഭിന്നകം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%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="परिमेय संख्या – Marathi" lang="mr" hreflang="mr" data-title="परिमेय 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_nisbah" title="Nombor nisbah – Malay" lang="ms" hreflang="ms" data-title="Nombor nisbah" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE" title="Рационал тоо – Mongolian" lang="mn" hreflang="mn" data-title="Рационал тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%90%E1%80%AD%E1%80%95%E1%80%BC%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="အတိပြကိန်း – Burmese" lang="my" hreflang="my" data-title="အတိပြကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rationaal_getal" title="Rationaal getal – Dutch" lang="nl" hreflang="nl" data-title="Rationaal getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="परिमेय सङ्ख्या – Nepali" lang="ne" hreflang="ne" data-title="परिमेय सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数 – Japanese" lang="ja" hreflang="ja" data-title="有理数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Ratsionaal_taal" title="Ratsionaal taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Ratsionaal taal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rasjonalt_tall" title="Rasjonalt tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Rasjonalt tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Rasjonale_tal" title="Rasjonale tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Rasjonale tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_racional" title="Nombre racional – Occitan" lang="oc" hreflang="oc" data-title="Nombre racional" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%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="ପରିମେୟ ସଂଖ୍ୟା – Odia" lang="or" hreflang="or" data-title="ପରିମେୟ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Entaa" title="Lakkoofsa Entaa – Oromo" lang="om" hreflang="om" data-title="Lakkoofsa Entaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ratsional_sonlar" title="Ratsional sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Ratsional sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%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="ਬਟੇਨੁਮਾ ਸੰਖਿਆ – Punjabi" lang="pa" hreflang="pa" data-title="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-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%B4%D9%86%D9%84_%D9%86%D9%85%D8%A8%D8%B1" title="ریشنل نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ریشنل نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Rashanal_nomba" title="Rashanal nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Rashanal nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_rassional" title="Nùmer rassional – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer rassional" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Ratschonale_Tall" title="Ratschonale Tall – Low German" lang="nds" hreflang="nds" data-title="Ratschonale Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_wymierne" title="Liczby wymierne – Polish" lang="pl" hreflang="pl" data-title="Liczby wymierne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_racional" title="Número racional – Portuguese" lang="pt" hreflang="pt" data-title="Número racional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Racional_sanlar" title="Racional sanlar – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Racional sanlar" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_ra%C8%9Bional" title="Număr rațional – Romanian" lang="ro" hreflang="ro" data-title="Număr rațional" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%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="Рациональное число – Russian" lang="ru" hreflang="ru" data-title="Рациональное число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_racional%C3%AB" title="Numrat racionalë – Albanian" lang="sq" hreflang="sq" data-title="Numrat racionalë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_razziunali" title="Nùmmuru razziunali – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru razziunali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B6%BB%E0%B7%92%E0%B6%B8%E0%B7%9A%E0%B6%BA_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="පරිමේය සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="පරිමේය සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Rational_number" title="Rational number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Rational 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/Racion%C3%A1lne_%C4%8D%C3%ADslo" title="Racionálne číslo – Slovak" lang="sk" hreflang="sk" data-title="Racionálne 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/Racionalno_%C5%A1tevilo" title="Racionalno število – Slovenian" lang="sl" hreflang="sl" data-title="Racionalno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%95%DB%8E%DA%98%DB%95%DB%8C%DB%8C" title="ژمارەی ڕێژەیی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی ڕێژەیی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%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="Рационалан број – Serbian" lang="sr" hreflang="sr" data-title="Рационалан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Racionalni_broj" title="Racionalni broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Racionalni broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rationaaliluku" title="Rationaaliluku – Finnish" lang="fi" hreflang="fi" data-title="Rationaaliluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rationella_tal" title="Rationella tal – Swedish" lang="sv" hreflang="sv" data-title="Rationella tal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Makatwirang_bilang" title="Makatwirang bilang – Tagalog" lang="tl" hreflang="tl" data-title="Makatwirang bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</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%AF%81_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="விகிதமுறு எண் – Tamil" lang="ta" hreflang="ta" data-title="விகிதமுறு எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C_%D1%81%D0%B0%D0%BD%D0%BD%D0%B0%D1%80" title="Рациональ саннар – Tatar" lang="tt" hreflang="tt" data-title="Рациональ саннар" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</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%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A2%E0%B8%B0" title="จำนวนตรรกยะ – Thai" lang="th" hreflang="th" data-title="จำนวนตรรกยะ" 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%D0%B8_%D1%80%D0%B0%D1%82%D1%81%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D3%A3" title="Адади ратсионалӣ – Tajik" lang="tg" hreflang="tg" data-title="Адади ратсионалӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" 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/Rasyonel_say%C4%B1lar" title="Rasyonel sayılar – Turkish" lang="tr" hreflang="tr" data-title="Rasyonel sayılar" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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%A0%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="Раціональне число – Ukrainian" lang="uk" hreflang="uk" data-title="Раціональне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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/%D9%86%D8%A7%D8%B7%D9%82_%D8%B9%D8%AF%D8%AF" title="ناطق عدد – Urdu" lang="ur" hreflang="ur" data-title="ناطق 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_h%E1%BB%AFu_t%E1%BB%89" title="Số hữu tỉ – Vietnamese" lang="vi" hreflang="vi" data-title="Số hữu tỉ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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/Jagoarv" title="Jagoarv – Võro" lang="vro" hreflang="vro" data-title="Jagoarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%88%86%E6%95%B8" title="分數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="分數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Rationoale_getalln" title="Rationoale getalln – West Flemish" lang="vls" hreflang="vls" data-title="Rationoale 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ihap_rasyonal" title="Ihap rasyonal – Waray" lang="war" hreflang="war" data-title="Ihap rasyonal" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数 – Wu" lang="wuu" hreflang="wuu" data-title="有理数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A8%D7%90%D7%A6%D7%99%D7%90%D7%A0%D7%90%D7%9C%D7%A2_%D7%A6%D7%90%D7%9C" title="ראציאנאלע צאל – Yiddish" lang="yi" hreflang="yi" data-title="ראציאנאלע צאל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</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_on%C3%AD%C3%ACp%C3%ADn" title="Nọ́mbà oníìpín – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà oníìpín" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E6%95%B8" title="有理數 – Cantonese" lang="yue" hreflang="yue" data-title="有理數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Raciuonalosis_skaitlios" title="Raciuonalosis skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Raciuonalosis skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数 – Chinese" lang="zh" hreflang="zh" data-title="有理数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_rasional" title="Lumur rasional – Iban" lang="iba" hreflang="iba" data-title="Lumur rasional" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target"><span>Jaku Iban</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/Q1244890#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Rationals" redirects here. For other uses, see <a href="/wiki/Rational_(disambiguation)" class="mw-disambig" title="Rational (disambiguation)">Rational (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Number-systems_(NZQRC).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Number-systems_%28NZQRC%29.svg/220px-Number-systems_%28NZQRC%29.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Number-systems_%28NZQRC%29.svg/330px-Number-systems_%28NZQRC%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Number-systems_%28NZQRC%29.svg/440px-Number-systems_%28NZQRC%29.svg.png 2x" data-file-width="1000" data-file-height="500" /></a><figcaption>The rational numbers <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> are included in the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>⁠</span>, which are included in the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>⁠</span>, while rationals include the <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>⁠</span>, which in turn include the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>⁠</span>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>rational number</b> is a <a href="/wiki/Number" title="Number">number</a> that can be expressed as the <a href="/wiki/Quotient" title="Quotient">quotient</a> or <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> <span class="nowrap">⁠<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 {\tfrac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b38d2684323653daafdd152b7e988594003897d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.663ex; height:3.676ex;" alt="{\displaystyle {\tfrac {p}{q}}}"></span>⁠</span> of two <a href="/wiki/Integer" title="Integer">integers</a>, a <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerator</a> <span class="texhtml mvar" style="font-style:italic;">p</span> and a non-zero <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> <span class="texhtml mvar" style="font-style:italic;">q</span>.<sup id="cite_ref-Rosen_1-0" class="reference"><a href="#cite_note-Rosen-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, <span class="nowrap">⁠<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 {\tfrac {3}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f2b824decf224d9a3143d4271666c7fba7ac83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{7}}}"></span>⁠</span> is a rational number, as is every integer (for example, <span class="nowrap"><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 -5={\tfrac {-5}{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo>−</mo> <mn>5</mn> </mrow> <mn>1</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -5={\tfrac {-5}{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef5aab2ced61ba1099051521167fbbe0d346e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.006ex; height:3.676ex;" alt="{\displaystyle -5={\tfrac {-5}{1}}}"></span>).</span> The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all rational numbers, also referred to as "<b>the rationals</b>",<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the <b>field of rationals</b><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or the <b>field of rational numbers</b> is usually denoted by boldface <span class="texhtml"><b>Q</b></span>, or <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}"></span>⁠</span> </p><p>A rational number is a <a href="/wiki/Real_number" title="Real number">real number</a>. The real numbers that are rational are those whose <a href="/wiki/Decimal_expansion" class="mw-redirect" title="Decimal expansion">decimal expansion</a> either terminates after a finite number of <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> (example: <span class="texhtml">3/4 = 0.75</span>), or eventually begins to <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeat</a> the same finite <a href="/wiki/Sequence" title="Sequence">sequence</a> of digits over and over (example: <span class="texhtml">9/44 = 0.20454545...</span>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This statement is true not only in <a href="/wiki/Decimal" title="Decimal">base 10</a>, but also in every other integer <a href="/wiki/Radix" title="Radix">base</a>, such as the <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> and <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> ones (see <a href="/wiki/Repeating_decimal#Extension_to_other_bases" title="Repeating decimal">Repeating decimal § Extension to other bases</a>). </p><p>A <a href="/wiki/Real_number" title="Real number">real number</a> that is not rational is called <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Irrational numbers include the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> <span class="nowrap">(<span class="nowrap">⁠<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>⁠</span>),</span> <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>, <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span>, and the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> (<span class="texhtml mvar" style="font-style:italic;">φ</span>). Since the set of rational numbers is <a href="/wiki/Countable_set" title="Countable set">countable</a>, and the set of real numbers is <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a>, <a href="/wiki/Almost_all" title="Almost all">almost all</a> real numbers are irrational.<sup id="cite_ref-Rosen_1-1" class="reference"><a href="#cite_note-Rosen-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Rational numbers can be <a href="/wiki/Mathematical_jargon#formally" class="mw-redirect" title="Mathematical jargon">formally</a> defined as <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of pairs of integers <span class="texhtml">(<i>p, q</i>)</span> with <span class="texhtml"><i>q</i> ≠ 0</span>, using the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> defined as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c444812cfda6e835c1878e46448955515bcd9c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.687ex; height:2.843ex;" alt="{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}"></span></dd></dl> <p>The fraction <span class="nowrap">⁠<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 {\tfrac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b38d2684323653daafdd152b7e988594003897d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.663ex; height:3.676ex;" alt="{\displaystyle {\tfrac {p}{q}}}"></span>⁠</span> then denotes the equivalence class of <span class="texhtml">(<i>p, q</i>)</span>.<sup id="cite_ref-:1_6-0" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Rational numbers together with <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> which contains the <a href="/wiki/Integer" title="Integer">integers</a>, and is contained in any field containing the integers. In other words, the field of rational numbers is a <a href="/wiki/Prime_field" class="mw-redirect" title="Prime field">prime field</a>, and a field has <a href="/wiki/Characteristic_zero" class="mw-redirect" title="Characteristic zero">characteristic zero</a> if and only if it contains the rational numbers as a subfield. Finite <a href="/wiki/Field_extension" title="Field extension">extensions</a> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> are called <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a>, and the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is the field of <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>.<sup id="cite_ref-Gilbert_7-0" class="reference"><a href="#cite_note-Gilbert-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the rational numbers form a <a href="/wiki/Dense_set" title="Dense set">dense subset</a> of the real numbers. The real numbers can be constructed from the rational numbers by <a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">completion</a>, using <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>, <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cuts</a>, or infinite <a href="/wiki/Decimal" title="Decimal">decimals</a> (see <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">Construction of the real numbers</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i>rational</i> in reference to the set <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> refers to the fact that a rational number represents a <i><a href="/wiki/Ratio" title="Ratio">ratio</a></i> of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective <i>rational</i> sometimes means that the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> are rational numbers. For example, a <a href="/wiki/Rational_point" title="Rational point">rational point</a> is a point with rational <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> (i.e., a point whose coordinates are rational numbers); a <i>rational matrix</i> is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> of rational numbers; a <i>rational polynomial</i> may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "<a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational expression</a>" and "<a href="/wiki/Rational_function" title="Rational function">rational function</a>" (a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a <a href="/wiki/Rational_curve" class="mw-redirect" title="Rational curve">rational curve</a> <i>is not</i> a curve defined over the rationals, but a curve which can be parameterized by rational functions. </p> <div class="mw-heading mw-heading3"><h3 id="Etymology">Etymology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=2" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although nowadays <i>rational numbers</i> are defined in terms of <i>ratios</i>, the term <i>rational</i> is not a <a href="/wiki/Morphological_derivation" title="Morphological derivation">derivation</a> of <i>ratio</i>. On the contrary, it is <i>ratio</i> that is derived from <i>rational</i>: the first use of <i>ratio</i> with its modern meaning was attested in English about 1660,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> while the use of <i>rational</i> for qualifying numbers appeared almost a century earlier, in 1570.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This meaning of <i>rational</i> came from the mathematical meaning of <i>irrational</i>, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἄλογος</span></span>)".<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>This unusual history originated in the fact that <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greeks</a> "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> So such lengths were <i>irrational</i>, in the sense of <i>illogical</i>, that is "not to be spoken about" (<span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἄλογος</span></span> in Greek).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetic">Arithmetic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=3" title="Edit section: Arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Fraction_(mathematics)#Arithmetic_with_fractions" class="mw-redirect" title="Fraction (mathematics)">Fraction (mathematics) § Arithmetic with fractions</a></div> <div class="mw-heading mw-heading3"><h3 id="Irreducible_fraction">Irreducible fraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=4" title="Edit section: Irreducible fraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every rational number may be expressed in a unique way as an <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">irreducible fraction</a> <span class="nowrap">⁠<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 {\tfrac {a}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62b49e759a120ce9f412ae2488478c38a87e5b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.353ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}},}"></span>⁠</span> where <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are <a href="/wiki/Coprime_integers" title="Coprime integers">coprime integers</a> and <span class="texhtml"><i>b</i> > 0</span>. This is often called the <a href="/wiki/Canonical_form" title="Canonical form">canonical form</a> of the rational number. </p><p>Starting from a rational number <span class="nowrap">⁠<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 {\tfrac {a}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62b49e759a120ce9f412ae2488478c38a87e5b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.353ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}},}"></span>⁠</span> its canonical form may be obtained by dividing <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> by their <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a>, and, if <span class="texhtml"><i>b</i> < 0</span>, changing the sign of the resulting numerator and denominator. </p> <div class="mw-heading mw-heading3"><h3 id="Embedding_of_integers">Embedding of integers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=5" title="Edit section: Embedding of integers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any integer <span class="texhtml mvar" style="font-style:italic;">n</span> can be expressed as the rational number <span class="nowrap">⁠<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 {\tfrac {n}{1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{1}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1150c1d89bf950938dfd5272820ea3136e22f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.469ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{1}},}"></span>⁠</span> which is its canonical form as a rational number. </p> <div class="mw-heading mw-heading3"><h3 id="Equality">Equality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=6" title="Edit section: Equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05bd90784f3c483ffcf3e46e198a25f63357a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.216ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}"></span> if and only if <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 ad=bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>d</mi> <mo>=</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ad=bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ab92607ba6422f676746108985227d9ad2091a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.549ex; height:2.176ex;" alt="{\displaystyle ad=bc}"></span></dd></dl> <p>If both fractions are in canonical form, then: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05bd90784f3c483ffcf3e46e198a25f63357a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.216ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}"></span> if and only if <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=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb3f1b1ad87e99791ba713839204a88b27239a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.676ex;" alt="{\displaystyle a=c}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d313b2b19adc53ca61b0af421d36cdb04e8507b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.312ex; height:2.176ex;" alt="{\displaystyle b=d}"></span><sup id="cite_ref-:1_6-1" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ordering">Ordering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=7" title="Edit section: Ordering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If both denominators are positive (particularly if both fractions are in canonical form): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}"> <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> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2a1c2dfed2e4b4b819fe015c24e839580c0385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.216ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}"></span> if and only if <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 ad<bc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>d</mi> <mo><</mo> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ad<bc.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbfcf64cb654cacb81669c3319868f83782b66a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.195ex; height:2.176ex;" alt="{\displaystyle ad<bc.}"></span></dd></dl> <p>On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator.<sup id="cite_ref-:1_6-2" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=8" title="Edit section: Addition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two fractions are added as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}"> <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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eecabb202924505dec89443299735337cd84d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.83ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}"></span></dd></dl> <p>If both fractions are in canonical form, the result is in canonical form if and only if <span class="texhtml mvar" style="font-style:italic;">b, d</span> are <a href="/wiki/Coprime_integers" title="Coprime integers">coprime integers</a>.<sup id="cite_ref-:1_6-3" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_14-0" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Subtraction">Subtraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=9" title="Edit section: Subtraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48ceb36618dae21637519ed938390f4d993b912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.83ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}"></span></dd></dl> <p>If both fractions are in canonical form, the result is in canonical form if and only if <span class="texhtml mvar" style="font-style:italic;">b, d</span> are <a href="/wiki/Coprime_integers" title="Coprime integers">coprime integers</a>.<sup id="cite_ref-:2_14-1" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=10" title="Edit section: Multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The rule for multiplication is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}"> <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> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54e5e6932095b006bbd8017a14a33973bdc33b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.615ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}"></span></dd></dl> <p>where the result may be a <a href="/wiki/Reducible_fraction" class="mw-redirect" title="Reducible fraction">reducible fraction</a>—even if both original fractions are in canonical form.<sup id="cite_ref-:1_6-4" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_14-2" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Inverse">Inverse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=11" title="Edit section: Inverse"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every rational number <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> has an <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a>, often called its <i>opposite</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c0d8e49dd78e5306ff7f822f2be64ccf8fdcce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.656ex; height:5.176ex;" alt="{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}"></span></dd></dl> <p>If <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> is in canonical form, the same is true for its opposite. </p><p>A nonzero rational number <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>, also called its <i>reciprocal</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d43d29378c994b0cbb80869ef79976a41bd500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.985ex; height:5.509ex;" alt="{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}"></span></dd></dl> <p>If <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> is in canonical form, then the canonical form of its reciprocal is either <span class="nowrap">⁠<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 {\tfrac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe57b48d5ecb0ffb15021231156bfab5eacdd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.706ex; height:3.509ex;" alt="{\displaystyle {\tfrac {b}{a}}}"></span>⁠</span> or <span class="nowrap">⁠<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 {\tfrac {-b}{-a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {-b}{-a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a618628089043f1077c83ee808700053748ff24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.631ex; height:3.676ex;" alt="{\displaystyle {\tfrac {-b}{-a}},}"></span>⁠</span> depending on the sign of <span class="texhtml mvar" style="font-style:italic;">a</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Division">Division</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=12" title="Edit section: Division"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml mvar" style="font-style:italic;">b, c, d</span> are nonzero, the division rule is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mstyle> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fcf07471454a815a850690e745373bc29aaa2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:10.703ex; height:10.176ex;" alt="{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}"></span></dd></dl> <p>Thus, dividing <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> by <span class="nowrap">⁠<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 {\tfrac {c}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a2119495c75fea0bacd25e35ee40dc3897dffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.696ex; height:3.343ex;" alt="{\displaystyle {\tfrac {c}{d}}}"></span>⁠</span> is equivalent to multiplying <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> by the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of <span class="nowrap">⁠<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 {\tfrac {c}{d}}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {c}{d}}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec79af8efcc6ec77fb799083085e9c3e363d864e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.988ex; height:3.343ex;" alt="{\displaystyle {\tfrac {c}{d}}:}"></span>⁠</span><sup id="cite_ref-:2_14-3" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066abede11b6d8f3eaa9d180ff8b4a006c7bd88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.824ex; height:5.509ex;" alt="{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Exponentiation_to_integer_power">Exponentiation to integer power</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=13" title="Edit section: Exponentiation to integer power"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml mvar" style="font-style:italic;">n</span> is a non-negative integer, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1faff2c20b0639c50f80824e2dc22cad7e3f013b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.09ex; height:5.343ex;" alt="{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}"></span></dd></dl> <p>The result is in canonical form if the same is true for <span class="nowrap">⁠<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 {\tfrac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b66abff3723f9c14e12a0f1e2763690db77058c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.353ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}.}"></span>⁠</span> In particular, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11702a7d9df5922c666c0ef7a66aa039d362768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.803ex; height:5.343ex;" alt="{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}"></span></dd></dl> <p>If <span class="texhtml"><i>a</i> ≠ 0</span>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41093a5034d5a9389d789f2155bd090d16cfdc07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.368ex; height:5.509ex;" alt="{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}"></span></dd></dl> <p>If <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> is in canonical form, the canonical form of the result is <span class="nowrap">⁠<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 {\tfrac {b^{n}}{a^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {b^{n}}{a^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70746b1038f7f88b9eb467db23b124cc49ca054c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.671ex; height:3.676ex;" alt="{\displaystyle {\tfrac {b^{n}}{a^{n}}}}"></span>⁠</span> if <span class="texhtml"><i>a</i> > 0</span> or <span class="texhtml mvar" style="font-style:italic;">n</span> is even. Otherwise, the canonical form of the result is <span class="nowrap">⁠<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 {\tfrac {-b^{n}}{-a^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934e4844c19e31669eb8a1a764c734a7fe9ac406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.596ex; height:4.009ex;" alt="{\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}"></span>⁠</span> </p> <div class="mw-heading mw-heading2"><h2 id="Continued_fraction_representation">Continued fraction representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=14" title="Edit section: Continued fraction representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fraction</a></div> <p>A <b>finite continued fraction</b> is an expression such as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋱</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9e201490c5e9b7dccf04f9bf4a40eee991950e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:27.632ex; height:18.009ex;" alt="{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">a<sub>n</sub></span> are integers. Every rational number <span class="nowrap">⁠<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 {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}"></span>⁠</span> can be represented as a finite continued fraction, whose <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> <span class="texhtml mvar" style="font-style:italic;">a<sub>n</sub></span> can be determined by applying the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> to <span class="texhtml">(<i>a, b</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_representations">Other representations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=15" title="Edit section: Other representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Common_fraction" class="mw-redirect" title="Common fraction">common fraction</a>: <span class="nowrap">⁠<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 {\tfrac {8}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {8}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021e2eae7b4ccb7172277ca7369147f4efc3db1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {8}{3}}}"></span>⁠</span></li> <li><a href="/wiki/Mixed_numeral" class="mw-redirect" title="Mixed numeral">mixed numeral</a>: <span class="nowrap">⁠<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{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5469d1a78fadb5c0e695fc519e3afc5b8240b0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.821ex; height:3.676ex;" alt="{\displaystyle 2{\tfrac {2}{3}}}"></span>⁠</span></li> <li><a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> using a <a href="/wiki/Vinculum_(symbol)" title="Vinculum (symbol)">vinculum</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 2.{\overline {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.{\overline {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f095e406fa2e567c13be2cffab9ad99ff28dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.087ex; height:2.843ex;" alt="{\displaystyle 2.{\overline {6}}}"></span></li> <li>repeating decimal using <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</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 2.(6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.(6)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac188b275220a577b12fdd8603106d3eedc0c837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.781ex; height:2.843ex;" alt="{\displaystyle 2.(6)}"></span></li> <li><a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> using traditional typography: <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+{\tfrac {1}{1+{\tfrac {1}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+{\tfrac {1}{1+{\tfrac {1}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab99c83963cd3f374f008fafb8d4692741c759d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.598ex; height:5.509ex;" alt="{\displaystyle 2+{\tfrac {1}{1+{\tfrac {1}{2}}}}}"></span></li> <li>continued fraction in abbreviated notation: <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;1,2]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [2;1,2]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1887ebaaed6f055311780a6b5c6eb3c2d43481e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [2;1,2]}"></span></li> <li><a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</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 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581ce754c1a38cfe27c64e58a638632d7fb12c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.159ex; height:3.676ex;" alt="{\displaystyle 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}"></span></li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic#Canonical_representation_of_a_positive_integer" title="Fundamental theorem of arithmetic">prime power decomposition</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 2^{3}\times 3^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>×</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}\times 3^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ca0d8ef1422319d1d657151d0a8e87e8e57308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.552ex; height:2.676ex;" alt="{\displaystyle 2^{3}\times 3^{-1}}"></span></li> <li><a href="/wiki/Quote_notation" class="mw-redirect" title="Quote notation">quote notation</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 3'6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mo>′</mo> </msup> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3'6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d39254dbfdee54bc695f55892fa1e4b1148cb296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.01ex; height:2.509ex;" alt="{\displaystyle 3'6}"></span></li></ul> <p>are different ways to represent the same rational value. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_construction">Formal construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=16" title="Edit section: Formal construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rational_Representation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Rational_Representation.svg/300px-Rational_Representation.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Rational_Representation.svg/450px-Rational_Representation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Rational_Representation.svg/600px-Rational_Representation.svg.png 2x" data-file-width="629" data-file-height="629" /></a><figcaption>A diagram showing a representation of the equivalent classes of pairs of integers</figcaption></figure> <p>The rational numbers may be built as <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of <a href="/wiki/Integer" title="Integer">integers</a>.<sup id="cite_ref-:1_6-5" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_14-4" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>More precisely, let <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d79cffb163db19f1277ddd053a2e233c278bc89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.242ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}"></span>⁠</span> be the set of the pairs <span class="texhtml">(<i>m, n</i>)</span> of integers such <span class="texhtml"><i>n</i> ≠ 0</span>. An <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> is defined on this set by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e9ba99775e3f75b381e050dc0dceb24f3317a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.602ex; height:2.843ex;" alt="{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}"></span><sup id="cite_ref-:1_6-6" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_14-5" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Addition and multiplication can be defined by the following rules: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca49c05f6cc6de96c5ffa8a6b6a9989210cbb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.028ex; height:2.843ex;" alt="{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14439a52170bda9827a8940beb7cee8600ff0c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.29ex; height:2.843ex;" alt="{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}"></span><sup id="cite_ref-:1_6-7" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <p>This equivalence relation is a <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a>, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is the defined as the <a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a> by this equivalence relation, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3a1ed833ecfeeadbbff09a2888091e2d3a3c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.472ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}"></span>⁠</span> equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> and produces its <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a>.)<sup id="cite_ref-:1_6-8" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The equivalence class of a pair <span class="texhtml">(<i>m, n</i>)</span> is denoted <span class="nowrap">⁠<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 {\tfrac {m}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {m}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eced668d03c131e22957a0ab8bf8e56fe5906d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.926ex; height:3.009ex;" alt="{\displaystyle {\tfrac {m}{n}}.}"></span>⁠</span> Two pairs <span class="texhtml">(<i>m</i><sub>1</sub>, <i>n</i><sub>1</sub>)</span> and <span class="texhtml">(<i>m</i><sub>2</sub>, <i>n</i><sub>2</sub>)</span> belong to the same equivalence class (that is are equivalent) if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6b852917408d0eb19e2a7610a77ce70eec0d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.832ex; height:2.009ex;" alt="{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}"></span></dd></dl> <p>This means that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4141c8183b703bed494d4a2b4960e69b02c9d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.96ex; height:5.009ex;" alt="{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}"></span></dd></dl> <p>if and only if<sup id="cite_ref-:1_6-9" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_14-6" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6b852917408d0eb19e2a7610a77ce70eec0d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.832ex; height:2.009ex;" alt="{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}"></span></dd></dl> <p>Every equivalence class <span class="nowrap">⁠<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 {\tfrac {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1413d4c83a94e754258e428fcd72d4fc6bcc2e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.009ex;" alt="{\displaystyle {\tfrac {m}{n}}}"></span>⁠</span> may be represented by infinitely many pairs, since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520ab75c19f8ac1b15a8590462c2539521cf5f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.42ex; height:5.343ex;" alt="{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}"></span></dd></dl> <p>Each equivalence class contains a unique <i><a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">canonical representative element</a></i>. The canonical representative is the unique pair <span class="texhtml">(<i>m, n</i>)</span> in the equivalence class such that <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>, and <span class="texhtml"><i>n</i> > 0</span>. It is called the <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">representation in lowest terms</a> of the rational number. </p><p>The integers may be considered to be rational numbers identifying the integer <span class="texhtml mvar" style="font-style:italic;">n</span> with the rational number <span class="nowrap">⁠<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 {\tfrac {n}{1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0a51ba7c4a4f8abd44967459eefee6ae21ca2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.469ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{1}}.}"></span>⁠</span> </p><p>A <a href="/wiki/Total_order" title="Total order">total order</a> may be defined on the rational numbers, that extends the natural order of the integers. One has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c564a2e7b7ee8756878e5d98ec12e05a43ae0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.96ex; height:5.009ex;" alt="{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}"></span></dd></dl> <p>If </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d84d6896d62ad3a2a0830d20f79d3c068af0fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:34.945ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=17" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> of all rational numbers, together with the addition and multiplication operations shown above, forms a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>.<sup id="cite_ref-:1_6-10" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> has no <a href="/wiki/Field_automorphism" class="mw-redirect" title="Field automorphism">field automorphism</a> other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.) </p><p><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is a <a href="/wiki/Prime_field" class="mw-redirect" title="Prime field">prime field</a>, which is a field that has no subfield other than itself.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The rationals are the smallest field with <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> zero. Every field of characteristic zero contains a unique subfield isomorphic to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}"></span>⁠</span> </p><p>With the order defined above, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a><sup id="cite_ref-:2_14-7" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}"></span>⁠</span> </p><p><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of the <a href="/wiki/Integer" title="Integer">integers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}"></span>⁠</span><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}"></span>⁠</span> i.e. the field of roots of rational polynomials, is the field of <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>. </p><p>The rationals are a <a href="/wiki/Densely_ordered" class="mw-redirect" title="Densely ordered">densely ordered</a> set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.<sup id="cite_ref-:1_6-11" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> For example, for any two fractions such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}"> <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> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2a1c2dfed2e4b4b819fe015c24e839580c0385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.216ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}"></span></dd></dl> <p>(where <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,d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7e58bcff2e2b515311524d5bf62b63f24afac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.247ex; height:2.509ex;" alt="{\displaystyle b,d}"></span> are positive), we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}"> <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> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/390412ffa31cb47f35a26be80ac14277c8ee2d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.875ex; height:5.343ex;" alt="{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}"></span></dd></dl> <p>Any <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> set which is countable, dense (in the above sense), and has no least or greatest element is <a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphic</a> to the rational numbers.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Countability">Countability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=18" title="Edit section: Countability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diagonal_argument.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Diagonal_argument.svg/200px-Diagonal_argument.svg.png" decoding="async" width="200" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Diagonal_argument.svg/300px-Diagonal_argument.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Diagonal_argument.svg/400px-Diagonal_argument.svg.png 2x" data-file-width="429" data-file-height="425" /></a><figcaption>Illustration of the countability of the positive rationals</figcaption></figure> <p>The set of all rational numbers is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a>, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a> as in a <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any <a href="/wiki/Division_by_zero" title="Division by zero">non-zero</a> number divided by itself will always equal one. </p><p>It is possible to generate all of the rational numbers without such redundancies: examples include the <a href="/wiki/Calkin%E2%80%93Wilf_tree" title="Calkin–Wilf tree">Calkin–Wilf tree</a> and <a href="/wiki/Stern%E2%80%93Brocot_tree" title="Stern–Brocot tree">Stern–Brocot tree</a>. </p><p>As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is a <a href="/wiki/Null_set" title="Null set">null set</a>, that is, <a href="/wiki/Almost_all" title="Almost all">almost all</a> real numbers are irrational, in the sense of <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Real_numbers_and_topological_properties">Real numbers and topological properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=19" title="Edit section: Real numbers and topological properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The rationals are a <a href="/wiki/Dense_set" title="Dense set">dense subset</a> of the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>; every real number has rational numbers arbitrarily close to it.<sup id="cite_ref-:1_6-12" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A related property is that rational numbers are the only numbers with <a href="/wiki/Finite_set" title="Finite set">finite</a> expansions as <a href="/wiki/Continued_fraction" title="Continued fraction">regular continued fractions</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>In the usual <a href="/wiki/Topology" title="Topology">topology</a> of the real numbers, the rationals are neither an <a href="/wiki/Open_set" title="Open set">open set</a> nor a <a href="/wiki/Closed_set" title="Closed set">closed set</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>By virtue of their order, the rationals carry an <a href="/wiki/Order_topology" title="Order topology">order topology</a>. The rational numbers, as a subspace of the real numbers, also carry a <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>. The rational numbers form a <a href="/wiki/Metric_space" title="Metric space">metric space</a> by using the <a href="/wiki/Absolute_difference" title="Absolute difference">absolute difference</a> metric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=|x-y|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=|x-y|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e52a0f0a6de665968d8e001b1c8796679e60e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.909ex; height:2.843ex;" alt="{\displaystyle d(x,y)=|x-y|,}"></span> and this yields a third topology on <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}"></span>⁠</span> All three topologies coincide and turn the rationals into a <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a>. The rational numbers are an important example of a space which is not <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>. The rationals are characterized topologically as the unique <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> <a href="/wiki/Topological_property" title="Topological property">metrizable space</a> without <a href="/wiki/Isolated_point" title="Isolated point">isolated points</a>. The space is also <a href="/wiki/Totally_disconnected_space" title="Totally disconnected space">totally disconnected</a>. The rational numbers do not form a <a href="/wiki/Completeness_(topology)" class="mw-redirect" title="Completeness (topology)">complete metric space</a>, and the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> are the completion of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> under the metric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=|x-y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=|x-y|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994ad8add8719c1d111342afeb970648ffff41f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle d(x,y)=|x-y|}"></span> above.<sup id="cite_ref-:2_14-8" class="reference"><a href="#cite_note-:2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="p-adic_numbers"><i>p</i>-adic numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=20" title="Edit section: p-adic numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic number</a></div> <p>In addition to the absolute value metric mentioned above, there are other metrics which turn <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> into a topological field: </p><p>Let <span class="texhtml mvar" style="font-style:italic;">p</span> be a <a href="/wiki/Prime_number" title="Prime number">prime number</a> and for any non-zero integer <span class="texhtml mvar" style="font-style:italic;">a</span>, let <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|_{p}=p_{-n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|_{p}=p_{-n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/018de379ae065be98746ec0e5aecb3e2132fbd53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.994ex; height:3.176ex;" alt="{\displaystyle |a|_{p}=p_{-n},}"></span> where <span class="texhtml mvar" style="font-style:italic;">p<sup>n</sup></span> is the highest power of <span class="texhtml mvar" style="font-style:italic;">p</span> <a href="/wiki/Divisor" title="Divisor">dividing</a> <span class="texhtml mvar" style="font-style:italic;">a</span>. </p><p>In addition set <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 |0|_{p}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|_{p}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1d5397ccca35ad2edd8646408aee241666695e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.423ex; height:3.176ex;" alt="{\displaystyle |0|_{p}=0.}"></span> For any rational number <span class="nowrap">⁠<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> <mo>,</mo> </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/101faa653872fb89cf12cbea8c0a99b1c507943a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}},}"></span>⁠</span> we set </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ad6485731d3d5e96ce8e14e0164e5d7b1a42c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.583ex; height:7.176ex;" alt="{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}"></span></dd></dl> <p>Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{p}(x,y)=|x-y|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{p}(x,y)=|x-y|_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5338fdfb334c39fa5d6d6db14cd43ff677cc06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.373ex; height:3.176ex;" alt="{\displaystyle d_{p}(x,y)=|x-y|_{p}}"></span></dd></dl> <p>defines a <a href="/wiki/Metric_space" title="Metric space">metric</a> on <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}"></span>⁠</span><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The metric space <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ,d_{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,d_{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5150dba67390218f05dfaa8717f5e2c081ba98b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.919ex; height:3.009ex;" alt="{\displaystyle (\mathbb {Q} ,d_{p})}"></span>⁠</span> is not complete, and its completion is the <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic number field</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4b0959c6687de4ffc0158e0b1d35d2137774d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}.}"></span>⁠</span> <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a> states that any non-trivial <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute value</a> on the rational numbers <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>⁠</span> is equivalent to either the usual real absolute value or a <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic</a> absolute value. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic rational</a></li> <li><a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">Floating point</a></li> <li><a href="/wiki/Ford_circle" title="Ford circle">Ford circles</a></li> <li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rational</a></li> <li><a href="/wiki/Glossary_of_arithmetic_and_diophantine_geometry#N" title="Glossary of arithmetic and diophantine geometry">Naive height—height of a rational number in lowest term</a></li> <li><a href="/wiki/Niven%27s_theorem" title="Niven's theorem">Niven's theorem</a></li> <li><a href="/wiki/Rational_data_type" title="Rational data type">Rational data type</a></li> <li><i><a href="/wiki/Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry" title="Divine Proportions: Rational Trigonometry to Universal Geometry">Divine Proportions: Rational Trigonometry to Universal Geometry</a></i></li></ul> <table style="margin:2em; border:2px solid silver; font-size:95%; border-collapse:collapse"> <tbody><tr> <td> <table style="margin:4px; border:2px solid silver"> <tbody><tr> <td> <table style="margin:1em"> <caption><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number systems</a> </caption> <tbody><tr> <td><a href="/wiki/Complex_number" title="Complex number">Complex</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {C} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Real_number" title="Real number">Real</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {R} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a class="mw-selflink selflink">Rational</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Integer" title="Integer">Integer</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {Z} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Natural_number" title="Natural number">Natural</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }"></span> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>: 0 </td></tr> <tr> <td><a href="/wiki/One" class="mw-redirect" title="One">One</a>: 1 </td></tr> <tr> <td><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> </td></tr> <tr> <td><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Fraction" title="Fraction">Fraction</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a> </td></tr> <tr> <td><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a> </td></tr> <tr> <td><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> </td> <td> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Irrational_number" title="Irrational number">Irrational</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic irrational</a> </td></tr> <tr> <td><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a> </td></tr> <tr> <td><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Rosen-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rosen_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rosen_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRosen2007" class="citation book cs1">Rosen, Kenneth (2007). <i>Discrete Mathematics and its Applications</i> (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-288008-3" title="Special:BookSources/978-0-07-288008-3"><bdi>978-0-07-288008-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+and+its+Applications&rft.place=New+York%2C+NY&rft.pages=105%2C+158-160&rft.edition=6th&rft.pub=McGraw-Hill&rft.date=2007&rft.isbn=978-0-07-288008-3&rft.aulast=Rosen&rft.aufirst=Kenneth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLass2009" class="citation book cs1">Lass, Harry (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WAY_AwAAQBAJ"><i>Elements of Pure and Applied Mathematics</i></a> (illustrated ed.). Courier Corporation. p. 382. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47186-0" title="Special:BookSources/978-0-486-47186-0"><bdi>978-0-486-47186-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Pure+and+Applied+Mathematics&rft.pages=382&rft.edition=illustrated&rft.pub=Courier+Corporation&rft.date=2009&rft.isbn=978-0-486-47186-0&rft.aulast=Lass&rft.aufirst=Harry&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWAY_AwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WAY_AwAAQBAJ&pg=PA382">Extract of page 382</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinson1996" class="citation book cs1">Robinson, Julia (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_33D84OENIAC"><i>The Collected Works of Julia Robinson</i></a>. American Mathematical Soc. p. 104. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0575-6" title="Special:BookSources/978-0-8218-0575-6"><bdi>978-0-8218-0575-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Collected+Works+of+Julia+Robinson&rft.pages=104&rft.pub=American+Mathematical+Soc&rft.date=1996&rft.isbn=978-0-8218-0575-6&rft.aulast=Robinson&rft.aufirst=Julia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_33D84OENIAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_33D84OENIAC&pg=PA104">Extract of page 104</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/rational-number">"Rational number"</a>. <i>Encyclopedia Britannica</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=Rational+number&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Frational-number&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RationalNumber.html">"Rational Number"</a>. <i>Wolfram MathWorld</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Rational+Number&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRationalNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-:1-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_6-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:1_6-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:1_6-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:1_6-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:1_6-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:1_6-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-:1_6-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-:1_6-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-:1_6-11"><sup><i><b>l</b></i></sup></a> <a href="#cite_ref-:1_6-12"><sup><i><b>m</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggs2002" class="citation book cs1">Biggs, Norman L. (2002). <i>Discrete Mathematics</i>. India: Oxford University Press. pp. 75–78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-871369-2" title="Special:BookSources/978-0-19-871369-2"><bdi>978-0-19-871369-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics&rft.place=India&rft.pages=75-78&rft.pub=Oxford+University+Press&rft.date=2002&rft.isbn=978-0-19-871369-2&rft.aulast=Biggs&rft.aufirst=Norman+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-Gilbert-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gilbert_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbertLinda2005" class="citation book cs1">Gilbert, Jimmie; <a href="/wiki/Linda_Gilbert_Saucier" title="Linda Gilbert Saucier">Linda, Gilbert</a> (2005). <i>Elements of Modern Algebra</i> (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-40264-X" title="Special:BookSources/0-534-40264-X"><bdi>0-534-40264-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Modern+Algebra&rft.place=Belmont%2C+CA&rft.pages=243-244&rft.edition=6th&rft.pub=Thomson+Brooks%2FCole&rft.date=2005&rft.isbn=0-534-40264-X&rft.aulast=Gilbert&rft.aufirst=Jimmie&rft.au=Linda%2C+Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Oxford English Dictionary</i> (2nd ed.). Oxford University Press. 1989.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+English+Dictionary&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1989&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> Entry <b>ratio</b>, <i>n.</i>, sense 2.a.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Oxford English Dictionary</i> (2nd ed.). Oxford University Press. 1989.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+English+Dictionary&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1989&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> Entry <b>rational</b>, <i>a. (adv.)</i> and <i>n.</i><sup>1</sup>, sense 5.a.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Oxford English Dictionary</i> (2nd ed.). Oxford University Press. 1989.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+English+Dictionary&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1989&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> Entry <b>irrational</b>, <i>a.</i> and <i>n.</i>, sense 3.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShor2017" class="citation web cs1"><a href="/wiki/Peter_Shor" title="Peter Shor">Shor, Peter</a> (2017-05-09). <a rel="nofollow" class="external text" href="https://english.stackexchange.com/questions/217956/does-rational-come-from-ratio-or-ratio-come-from-rational/218079#218079">"Does rational come from ratio or ratio come from rational"</a>. <i>Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-03-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Exchange&rft.atitle=Does+rational+come+from+ratio+or+ratio+come+from+rational&rft.date=2017-05-09&rft.aulast=Shor&rft.aufirst=Peter&rft_id=https%3A%2F%2Fenglish.stackexchange.com%2Fquestions%2F217956%2Fdoes-rational-come-from-ratio-or-ratio-come-from-rational%2F218079%23218079&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoolman2016" class="citation web cs1">Coolman, Robert (2016-01-29). <a rel="nofollow" class="external text" href="https://nautil.us/blog/how-a-mathematical-superstition-stultified-algebra-for-over-a-thousand-years">"How a Mathematical Superstition Stultified Algebra for Over a Thousand Years"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-03-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+a+Mathematical+Superstition+Stultified+Algebra+for+Over+a+Thousand+Years&rft.date=2016-01-29&rft.aulast=Coolman&rft.aufirst=Robert&rft_id=https%3A%2F%2Fnautil.us%2Fblog%2Fhow-a-mathematical-superstition-stultified-algebra-for-over-a-thousand-years&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKramer1983" class="citation book cs1">Kramer, Edna (1983). <i>The Nature and Growth of Modern Mathematics</i>. Princeton University Press. p. 28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+and+Growth+of+Modern+Mathematics&rft.pages=28&rft.pub=Princeton+University+Press&rft.date=1983&rft.aulast=Kramer&rft.aufirst=Edna&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-:2-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:2_14-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:2_14-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:2_14-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:2_14-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:2_14-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:2_14-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:2_14-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Fraction">"Fraction - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. 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London, England: MIT Press. p. 578. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-2625-9020-4" title="Special:BookSources/0-2625-9020-4"><bdi>0-2625-9020-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedic+Dictionary+of+Mathematics%2C+Volume+1&rft.place=London%2C+England&rft.pages=578&rft.pub=MIT+Press&rft.date=1993&rft.isbn=0-2625-9020-4&rft.aulast=S%C5%ABgakkai&rft.aufirst=Nihon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWHjO9K6xEm4C%26pg%3DPA578&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki2003" class="citation book cs1">Bourbaki, N. (2003). <i>Algebra II: Chapters 4 - 7</i>. Springer Science & Business Media. p. A.VII.5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+II%3A+Chapters+4+-+7&rft.pages=A.VII.5&rft.pub=Springer+Science+%26+Business+Media&rft.date=2003&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGieseSchönegge1995" class="citation techreport cs1">Giese, Martin; Schönegge, Arno (December 1995). <a rel="nofollow" class="external text" href="https://www.uio.no/studier/emner/matnat/ifi/nedlagte-emner/INF5170/v14/undervisningsmateriale/countable-densely-ordered-sets.pdf"><i>Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV</i></a> <span class="cs1-format">(PDF)</span> (Technical report)<span class="reference-accessdate">. Retrieved <span class="nowrap">17 August</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Any+two+countable+densely+ordered+sets+without+endpoints+are+isomorphic+-+a+formal+proof+with+KIV&rft.date=1995-12&rft.aulast=Giese&rft.aufirst=Martin&rft.au=Sch%C3%B6negge%2C+Arno&rft_id=https%3A%2F%2Fwww.uio.no%2Fstudier%2Femner%2Fmatnat%2Fifi%2Fnedlagte-emner%2FINF5170%2Fv14%2Fundervisningsmateriale%2Fcountable-densely-ordered-sets.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnthony_VazzanaDavid_Garth2015" class="citation book cs1">Anthony Vazzana; David Garth (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iJWwDwAAQBAJ"><i>Introduction to Number Theory</i></a> (2nd, revised ed.). CRC Press. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-1752-6" title="Special:BookSources/978-1-4987-1752-6"><bdi>978-1-4987-1752-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Number+Theory&rft.pages=1&rft.edition=2nd%2C+revised&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-4987-1752-6&rft.au=Anthony+Vazzana&rft.au=David+Garth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiJWwDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iJWwDwAAQBAJ&pg=RA1">Extract of page 1</a></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_A._Holmgren2012" class="citation book cs1">Richard A. Holmgren (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5LUPBwAAQBAJ"><i>A First Course in Discrete Dynamical Systems</i></a> (2nd, illustrated ed.). Springer Science & Business Media. p. 26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-8732-7" title="Special:BookSources/978-1-4419-8732-7"><bdi>978-1-4419-8732-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Discrete+Dynamical+Systems&rft.pages=26&rft.edition=2nd%2C+illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4419-8732-7&rft.au=Richard+A.+Holmgren&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5LUPBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5LUPBwAAQBAJ&pg=PA26">Extract of page 26</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/p-adicNumber.html">"p-adic Number"</a>. <i>Wolfram MathWorld</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-08-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=p-adic+Number&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2Fp-adicNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rational_number&action=edit&section=23" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-lower-alpha"> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span 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typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Rational_numbers" class="extiw" title="commons:Category:Rational numbers">Rational numbers</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Rational_numbers" class="extiw" title="v:Rational numbers">Rational numbers</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Rational_number">"Rational number"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rational+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DRational_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARational+number" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RationalNumber.html">"Rational Number" From MathWorld – A Wolfram Web Resource</a></li></ul> <div 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title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Square_root_of_6" title="Square root of 6">Square root of 6</a></li> <li><a href="/wiki/Square_root_of_7" title="Square root of 7">Square root of 7</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (<span class="texhtml mvar" style="font-style:italic;">ψ</span>)</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (<span class="texhtml mvar" style="font-style:italic;">ς</span>)</a></li> <li><a href="/wiki/Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" 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class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a class="mw-selflink selflink">Rational numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Rational_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Rational_numbers" title="Template:Rational numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Rational_numbers" title="Template talk:Rational numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Rational_numbers" title="Special:EditPage/Template:Rational numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Rational_numbers" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Rational numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integer" title="Integer">Integer</a></li> <li><a href="/wiki/Dedekind_cut" title="Dedekind 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