Repeating decimal - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Repeating decimal - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"41034a7e-cf2e-4ec2-ad67-01d3a4c62dff","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Repeating_decimal","wgTitle":"Repeating decimal","wgCurRevisionId":1258671238,"wgRevisionId":1258671238,"wgArticleId":13612447,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Articles needing additional references from July 2023","All articles needing additional references","Articles needing cleanup from July 2024","All pages needing cleanup","Cleanup tagged articles with a reason field from July 2024","Wikipedia pages needing cleanup from July 2024","Articles with multiple maintenance issues","Articles needing additional references from May 2022", "All articles with unsourced statements","Articles with unsourced statements from October 2023","Elementary arithmetic","Numeral systems"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Repeating_decimal","wgRelevantArticleId":13612447,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":60000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage", "wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q389208","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","ext.pygments":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.tablesorter.styles":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready", "wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.pygments.view","mediawiki.page.media","site","mediawiki.page.ready","jquery.tablesorter","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.pygments%2CwikimediaBadges%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cjquery.tablesorter.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Repeating decimal - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Repeating_decimal"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Repeating_decimal&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Repeating_decimal"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Repeating_decimal rootpage-Repeating_decimal skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Repeating+decimal" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Repeating+decimal" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Repeating+decimal" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Repeating+decimal" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> 1 Background </div> </a> <button aria-controls="toc-Background-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Background subsection </button> <ul id="toc-Background-sublist" class="vector-toc-list"> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> 1.1 Notation </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decimal_expansion_and_recurrence_sequence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decimal_expansion_and_recurrence_sequence"> <div class="vector-toc-text"> 1.2 Decimal expansion and recurrence sequence </div> </a> <ul id="toc-Decimal_expansion_and_recurrence_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Every_rational_number_is_either_a_terminating_or_repeating_decimal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Every_rational_number_is_either_a_terminating_or_repeating_decimal"> <div class="vector-toc-text"> 1.3 Every rational number is either a terminating or repeating decimal </div> </a> <ul id="toc-Every_rational_number_is_either_a_terminating_or_repeating_decimal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Every_repeating_or_terminating_decimal_is_a_rational_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Every_repeating_or_terminating_decimal_is_a_rational_number"> <div class="vector-toc-text"> 1.4 Every repeating or terminating decimal is a rational number </div> </a> <ul id="toc-Every_repeating_or_terminating_decimal_is_a_rational_number-sublist" class="vector-toc-list"> <li id="toc-Formal_proof" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formal_proof"> <div class="vector-toc-text"> 1.4.1 Formal proof </div> </a> <ul id="toc-Formal_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Table_of_values" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Table_of_values"> <div class="vector-toc-text"> 2 Table of values </div> </a> <ul id="toc-Table_of_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractions_with_prime_denominators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fractions_with_prime_denominators"> <div class="vector-toc-text"> 3 Fractions with prime denominators </div> </a> <button aria-controls="toc-Fractions_with_prime_denominators-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Fractions with prime denominators subsection </button> <ul id="toc-Fractions_with_prime_denominators-sublist" class="vector-toc-list"> <li id="toc-Cyclic_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_numbers"> <div class="vector-toc-text"> 3.1 Cyclic numbers </div> </a> <ul id="toc-Cyclic_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_reciprocals_of_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_reciprocals_of_primes"> <div class="vector-toc-text"> 3.2 Other reciprocals of primes </div> </a> <ul id="toc-Other_reciprocals_of_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Totient_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Totient_rule"> <div class="vector-toc-text"> 3.3 Totient rule </div> </a> <ul id="toc-Totient_rule-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Reciprocals_of_composite_integers_coprime_to_10" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reciprocals_of_composite_integers_coprime_to_10"> <div class="vector-toc-text"> 4 Reciprocals of composite integers coprime to 10 </div> </a> <ul id="toc-Reciprocals_of_composite_integers_coprime_to_10-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocals_of_integers_not_coprime_to_10" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reciprocals_of_integers_not_coprime_to_10"> <div class="vector-toc-text"> 5 Reciprocals of integers not coprime to 10 </div> </a> <ul id="toc-Reciprocals_of_integers_not_coprime_to_10-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Converting_repeating_decimals_to_fractions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Converting_repeating_decimals_to_fractions"> <div class="vector-toc-text"> 6 Converting repeating decimals to fractions </div> </a> <button aria-controls="toc-Converting_repeating_decimals_to_fractions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Converting repeating decimals to fractions subsection </button> <ul id="toc-Converting_repeating_decimals_to_fractions-sublist" class="vector-toc-list"> <li id="toc-A_shortcut" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_shortcut"> <div class="vector-toc-text"> 6.1 A shortcut </div> </a> <ul id="toc-A_shortcut-sublist" class="vector-toc-list"> <li id="toc-In_compressed_form" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#In_compressed_form"> <div class="vector-toc-text"> 6.1.1 In compressed form </div> </a> <ul id="toc-In_compressed_form-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Repeating_decimals_as_infinite_series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Repeating_decimals_as_infinite_series"> <div class="vector-toc-text"> 7 Repeating decimals as infinite series </div> </a> <ul id="toc-Repeating_decimals_as_infinite_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication_and_cyclic_permutation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multiplication_and_cyclic_permutation"> <div class="vector-toc-text"> 8 Multiplication and cyclic permutation </div> </a> <ul id="toc-Multiplication_and_cyclic_permutation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_properties_of_repetend_lengths" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_properties_of_repetend_lengths"> <div class="vector-toc-text"> 9 Other properties of repetend lengths </div> </a> <ul id="toc-Other_properties_of_repetend_lengths-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_to_other_bases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_to_other_bases"> <div class="vector-toc-text"> 10 Extension to other bases </div> </a> <button aria-controls="toc-Extension_to_other_bases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extension to other bases subsection </button> <ul id="toc-Extension_to_other_bases-sublist" class="vector-toc-list"> <li id="toc-Algorithm_for_positive_bases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algorithm_for_positive_bases"> <div class="vector-toc-text"> 10.1 Algorithm for positive bases </div> </a> <ul id="toc-Algorithm_for_positive_bases-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_to_cryptography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_to_cryptography"> <div class="vector-toc-text"> 11 Applications to cryptography </div> </a> <ul id="toc-Applications_to_cryptography-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"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Repeating decimal</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 33 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-33" aria-hidden="true" > 33 languages </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/Repeterende_breuk" title="Repeterende breuk – Afrikaans" lang="af" hreflang="af" data-title="Repeterende breuk" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%83%D8%B1%D8%A7%D8%B1_%D8%B9%D8%B4%D8%B1%D9%8A" title="تكرار عشري – Arabic" lang="ar" hreflang="ar" data-title="تكرار عشري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_decimal_peri%C3%B2dic" title="Nombre decimal periòdic – Catalan" lang="ca" hreflang="ca" data-title="Nombre decimal periòdic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Perioda_(matematika)" title="Perioda (matematika) – Czech" lang="cs" hreflang="cs" data-title="Perioda (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rationale_Zahl#Dezimalbruchentwicklung" 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">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%BF%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Περιοδικός αριθμός – Greek" lang="el" hreflang="el" data-title="Περιοδικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_decimal_peri%C3%B3dico" title="Número decimal periódico – Spanish" lang="es" hreflang="es" data-title="Número decimal periódico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_hamartar_periodiko" title="Zenbaki hamartar periodiko – Basque" lang="eu" hreflang="eu" data-title="Zenbaki hamartar periodiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%87%E2%80%8C%D8%AF%D9%87%DB%8C_%D9%85%D8%AA%D9%86%D8%A7%D9%88%D8%A8" title="ده‌دهی متناوب – Persian" lang="fa" hreflang="fa" data-title="ده‌دهی متناوب" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique" title="Développement décimal périodique – French" lang="fr" hreflang="fr" data-title="Développement décimal périodique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Decimal_peri%C3%B3dico" title="Decimal periódico – Galician" lang="gl" hreflang="gl" data-title="Decimal periódico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%9C%ED%99%98%EC%86%8C%EC%88%98" title="순환소수 – Korean" lang="ko" hreflang="ko" data-title="순환소수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_decimale_periodico" title="Numero decimale periodico – Italian" lang="it" hreflang="it" data-title="Numero decimale periodico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%91%D7%A8_%D7%9E%D7%97%D7%96%D7%95%D7%A8%D7%99" title="שבר מחזורי – Hebrew" lang="he" hreflang="he" data-title="שבר מחזורי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/De%C4%8Bimali_ripetuti" title="Deċimali ripetuti – Maltese" lang="mt" hreflang="mt" data-title="Deċimali ripetuti" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Repeterende_breuk" title="Repeterende breuk – Dutch" lang="nl" hreflang="nl" data-title="Repeterende breuk" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AA%E7%92%B0%E5%B0%8F%E6%95%B0" title="循環小数 – Japanese" lang="ja" hreflang="ja" data-title="循環小数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Periodisk_desimaltall" title="Periodisk desimaltall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Periodisk desimaltall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BE%E0%A8%B0%E0%A8%B5%E0%A8%BE%E0%A8%B0%E0%A9%80_%E0%A8%87%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%B0%E0%A9%80%E0%A8%86" title="ਵਾਰਵਾਰੀ ਇਸ਼ਾਰੀਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਵਾਰਵਾਰੀ ਇਸ਼ਾਰੀਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/U%C5%82amek_dziesi%C4%99tny_niesko%C5%84czony" title="Ułamek dziesiętny nieskończony – Polish" lang="pl" hreflang="pl" data-title="Ułamek dziesiętny nieskończony" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/D%C3%ADzima_peri%C3%B3dica" title="Dízima periódica – Portuguese" lang="pt" hreflang="pt" data-title="Dízima periódica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B8%D0%BE%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B4%D0%B5%D1%81%D1%8F%D1%82%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D0%B4%D1%80%D0%BE%D0%B1%D1%8C" title="Периодическая десятичная дробь – Russian" lang="ru" hreflang="ru" data-title="Периодическая десятичная дробь" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Repeating_decimal" title="Repeating decimal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Repeating decimal" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Peri%C3%B3da_(matematika)" title="Perióda (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Perióda (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Deseti%C5%A1ki_ulomek" title="Desetiški ulomek – Slovenian" lang="sl" hreflang="sl" data-title="Desetiški ulomek" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Jaksollinen_desimaaliluku" title="Jaksollinen desimaaliluku – Finnish" lang="fi" hreflang="fi" data-title="Jaksollinen desimaaliluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%80%E0%AE%B3%E0%AF%81%E0%AE%AE%E0%AF%8D_%E0%AE%A4%E0%AE%9A%E0%AE%AE%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%8D" title="மீளும் தசமங்கள் – Tamil" lang="ta" hreflang="ta" data-title="மீளும் தசமங்கள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A8%E0%B8%99%E0%B8%B4%E0%B8%A2%E0%B8%A1%E0%B8%8B%E0%B9%89%E0%B8%B3" title="ทศนิยมซ้ำ – Thai" lang="th" hreflang="th" data-title="ทศนิยมซ้ำ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Devirli_say%C4%B1" title="Devirli sayı – Turkish" lang="tr" hreflang="tr" data-title="Devirli sayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%84%D8%A7%D9%85%D8%B9%DB%8C%D8%AF_%D8%A7%D8%B9%D8%B4%D8%A7%D8%B1%DB%8C%DB%81" title="لامعید اعشاریہ – Urdu" lang="ur" hreflang="ur" data-title="لامعید اعشاریہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_th%E1%BA%ADp_ph%C3%A2n_v%C3%B4_h%E1%BA%A1n_tu%E1%BA%A7n_ho%C3%A0n" title="Số thập phân vô hạn tuần hoàn – Vietnamese" lang="vi" hreflang="vi" data-title="Số thập phân vô hạn tuần hoàn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%AA%E7%92%B0%E5%B0%8F%E6%95%B8" title="循環小數 – Cantonese" lang="yue" hreflang="yue" data-title="循環小數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BE%AA%E7%8E%AF%E5%B0%8F%E6%95%B0" title="循环小数 – Chinese" lang="zh" hreflang="zh" data-title="循环小数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q389208#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></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 class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Repeating_decimal" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Repeating_decimal" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Repeating_decimal">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Repeating_decimal&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Repeating_decimal&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Repeating_decimal">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Repeating_decimal&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Repeating_decimal&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Repeating_decimal" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Repeating_decimal" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Repeating_decimal&oldid=1258671238" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Repeating_decimal&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Repeating_decimal&id=1258671238&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FRepeating_decimal">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FRepeating_decimal">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Repeating_decimal&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Repeating_decimal&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q389208" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Decimal representation of a number whose digits are periodic</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Repeating fraction" redirects here. Not to be confused with <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1248332772">.mw-parser-output .multiple-issues-text{width:95%;margin:0.2em 0}.mw-parser-output .multiple-issues-text>.mw-collapsible-content{margin-top:0.3em}.mw-parser-output .compact-ambox .ambox{border:none;border-collapse:collapse;background-color:transparent;margin:0 0 0 1.6em!important;padding:0!important;width:auto;display:block}body.mediawiki .mw-parser-output .compact-ambox .ambox.mbox-small-left{font-size:100%;width:auto;margin:0}.mw-parser-output .compact-ambox .ambox .mbox-text{padding:0!important;margin:0!important}.mw-parser-output .compact-ambox .ambox .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:disc}body.skin-minerva .mw-parser-output .multiple-issues-text>.mw-collapsible-toggle,.mw-parser-output .compact-ambox .ambox .mbox-image,.mw-parser-output .compact-ambox .ambox .mbox-imageright,.mw-parser-output .compact-ambox .ambox .mbox-empty-cell,.mw-parser-output .compact-ambox .hide-when-compact{display:none}</style><table class="box-Multiple_issues plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible">This article has multiple issues. Please help <a href="/wiki/Special:EditPage/Repeating_decimal" title="Special:EditPage/Repeating decimal">improve it</a> or discuss these issues on the <a href="/wiki/Talk:Repeating_decimal" title="Talk:Repeating decimal">talk page</a>. (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove these messages</a>) <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This article needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>. Please help <a href="/wiki/Special:EditPage/Repeating_decimal" title="Special:EditPage/Repeating decimal">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed. Find sources: <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Repeating+decimal%22">"Repeating decimal"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Repeating+decimal%22+-wikipedia&tbs=ar:1">news</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Repeating+decimal%22&tbs=bkt:s&tbm=bks">newspapers</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Repeating+decimal%22+-wikipedia">books</a> · <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Repeating+decimal%22">scholar</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Repeating+decimal%22&acc=on&wc=on">JSTOR</a> (July 2023) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article may require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: The article is polluted with multiple non-encyclopedic tables, often placed before the true content that is hidden by them. Section are placed in a random order with the most important facts relegated toward the end of the article. Please help <a href="/wiki/Special:EditPage/Repeating_decimal" title="Special:EditPage/Repeating decimal">improve this article</a> if you can. (July 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> </div> </div> (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> A repeating decimal or recurring decimal is a <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a> of a number whose <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> are eventually <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is <a href="/wiki/Rational_number" title="Rational number">rational</a> if and only if its decimal representation is repeating or terminating. For example, the decimal representation of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/3⁠ becomes periodic just after the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a>, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3227/555⁠, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠593/53⁠, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830.... The infinitely repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zeros can be omitted and the decimal terminates before these zeros.<a href="#cite_note-1">[1]</a> Every terminating decimal representation can be written as a <a href="/wiki/Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fraction</a>, a fraction whose denominator is a <a href="/wiki/Power_(math)" class="mw-redirect" title="Power (math)">power</a> of 10 (e.g. 1.585 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1585/1000⁠); it may also be written as a <a href="/wiki/Ratio" title="Ratio">ratio</a> of the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠k/2n·5m⁠ (e.g. 1.585 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠317/23·52⁠). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9. This is obtained by decreasing the final (rightmost) non-zero digit by one and appending a repetend of 9. Two examples of this are <a href="/wiki/0.999..." title="0.999...">1.000... = 0.999...</a> and 1.585000... = 1.584999.... (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual <a href="/wiki/Division_algorithm" title="Division algorithm">division algorithm</a>.<a href="#cite_note-2">[2]</a>) Any number that cannot be expressed as a <a href="/wiki/Ratio" title="Ratio">ratio</a> of two <a href="/wiki/Integer" title="Integer">integers</a> is said to be <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>. Their decimal representation neither terminates nor infinitely repeats, but extends forever without repetition (see <a href="#Every_rational_number_is_either_a_terminating_or_repeating_decimal">§ Every rational number is either a terminating or repeating decimal</a>). Examples of such irrational numbers are <a href="/wiki/Square_root_of_2" title="Square root of 2">√2</a> and <a href="/wiki/Pi" title="Pi">π</a>.<a href="#cite_note-3">[3]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=1" title="Edit section: Background">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=2" title="Edit section: Notation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Repeating_decimal" title="Special:EditPage/Repeating decimal">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2022) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> There are several notational conventions for representing repeating decimals. None of them are accepted universally. <table class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"> <caption>Different notations with examples </caption> <tbody><tr> <th colspan="2">Fraction </th> <th><a href="/wiki/Vinculum_(symbol)" title="Vinculum (symbol)">Vinculum</a> </th> <th>Dots </th> <th><a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">Parentheses</a> </th> <th>Arc </th> <th><a href="/wiki/Ellipsis" title="Ellipsis">Ellipsis</a> </th></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/9⁠</td> <td style="padding:0;border-left:none;"> </td> <td>0.1 </td> <td>0..1 </td> <td>0.(1) </td> <td>0.1 </td> <td>0.111... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/9⁠ </td> <td>0.3 </td> <td>0..3 </td> <td>0.(3) </td> <td>0.3 </td> <td>0.333... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/3⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/9⁠ </td> <td>0.6 </td> <td>0..6 </td> <td>0.(6) </td> <td>0.6 </td> <td>0.666... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/11⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/99⁠ </td> <td>0.81 </td> <td>0..8.1 </td> <td>0.(81) </td> <td>0.81 </td> <td>0.8181... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/12⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠525/900⁠ </td> <td>0.583 </td> <td>0.58.3 </td> <td>0.58(3) </td> <td>0.583 </td> <td>0.58333... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠142857/999999⁠ </td> <td>0.142857 </td> <td>0..14285.7 </td> <td>0.(142857) </td> <td>0.142857 </td> <td>0.142857142857... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/81⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12345679/999999999⁠ </td> <td>0.012345679 </td> <td>0..01234567.9 </td> <td>0.(012345679) </td> <td>0.012345679 </td> <td>0.012345679012345679... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠22/7⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3142854/999999⁠ </td> <td>3.142857 </td> <td>3..14285.7 </td> <td>3.(142857) </td> <td>3.142857 </td> <td>3.142857142857... </td></tr> <tr> <td style="text-align:center;border-right:none;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠593/53⁠</td> <td style="padding:0;border-left:none;">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠111886792452819/9999999999999⁠ </td> <td>11.1886792452830 </td> <td>11..188679245283.0 </td> <td>11.(1886792452830) </td> <td>11.1886792452830 </td> <td>11.18867924528301886792452830... </td></tr></tbody></table> <ul><li>Vinculum: In the <a href="/wiki/United_States" title="United States">United States</a>, <a href="/wiki/Canada" title="Canada">Canada</a>, <a href="/wiki/India" title="India">India</a>, <a href="/wiki/France" title="France">France</a>, <a href="/wiki/Germany" title="Germany">Germany</a>, <a href="/wiki/Italy" title="Italy">Italy</a>, <a href="/wiki/Switzerland" title="Switzerland">Switzerland</a>, the <a href="/wiki/Czech_Republic" title="Czech Republic">Czech Republic</a>, <a href="/wiki/Slovakia" title="Slovakia">Slovakia</a>, <a href="/wiki/Slovenia" title="Slovenia">Slovenia</a>, <a href="/wiki/Chile" title="Chile">Chile</a>, and <a href="/wiki/Turkey" title="Turkey">Turkey</a>, the convention is to draw a horizontal line (a vinculum) above the repetend.<a href="#cite_note-4">[4]</a></li> <li>Dots: In some Islamic countries, such as <a href="/wiki/Malaysia" title="Malaysia">Malaysia</a>, <a href="/wiki/Morocco" title="Morocco">Morocco</a>, <a href="/wiki/Pakistan" title="Pakistan">Pakistan</a>, <a href="/wiki/Tunisia" title="Tunisia">Tunisia</a>, <a href="/wiki/Iran" title="Iran">Iran</a>, <a href="/wiki/Algeria" title="Algeria">Algeria</a> and <a href="/wiki/Egypt" title="Egypt">Egypt</a>, as well as the <a href="/wiki/United_Kingdom" title="United Kingdom">United Kingdom</a>, <a href="/wiki/New_Zealand" title="New Zealand">New Zealand</a>, <a href="/wiki/Australia" title="Australia">Australia</a>, <a href="/wiki/South_Africa" title="South Africa">South Africa</a>, <a href="/wiki/Japan" title="Japan">Japan</a>, <a href="/wiki/Thailand" title="Thailand">Thailand</a>, India, <a href="/wiki/South_Korea" title="South Korea">South Korea</a>, <a href="/wiki/Singapore" title="Singapore">Singapore</a>, and the <a href="/wiki/People%27s_Republic_of_China" class="mw-redirect" title="People's Republic of China">People's Republic of China</a>, the convention is to place dots above the outermost numerals of the repetend.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li> <li>Parentheses: In parts of <a href="/wiki/Europe" title="Europe">Europe</a>, incl. <a href="/wiki/Austria" title="Austria">Austria</a>, <a href="/wiki/Denmark" title="Denmark">Denmark</a>, <a href="/wiki/Finland" title="Finland">Finland</a>, the <a href="/wiki/Netherlands" title="Netherlands">Netherlands</a>, <a href="/wiki/Norway" title="Norway">Norway</a>, <a href="/wiki/Poland" title="Poland">Poland</a>, <a href="/wiki/Russia" title="Russia">Russia</a> and <a href="/wiki/Ukraine" title="Ukraine">Ukraine</a>, as well as <a href="/wiki/Vietnam" title="Vietnam">Vietnam</a> and <a href="/wiki/Israel" title="Israel">Israel</a>, the convention is to enclose the repetend in parentheses. This can cause confusion with the notation for <a href="/wiki/Standard_uncertainty" class="mw-redirect" title="Standard uncertainty">standard uncertainty</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li> <li>Arc: In <a href="/wiki/Spain" title="Spain">Spain</a> and some <a href="/wiki/Latin_America" title="Latin America">Latin American</a> countries, such as <a href="/wiki/Argentina" title="Argentina">Argentina</a>, <a href="/wiki/Brazil" title="Brazil">Brazil</a>, and <a href="/wiki/Mexico" title="Mexico">Mexico</a>, the arc notation over the repetend is also used as an alternative to the vinculum and the dots notation.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li> <li>Ellipsis: Informally, repeating decimals are often represented by an ellipsis (three periods, 0.333...), especially when the previous notational conventions are first taught in school. This notation introduces uncertainty as to which digits should be repeated and even whether repetition is occurring at all, since such ellipses are also employed for <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>; <a href="/wiki/Pi" title="Pi">π</a>, for example, can be represented as 3.14159....[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li></ul> In English, there are various ways to read repeating decimals aloud. For example, 1.234 may be read "one point two repeating three four", "one point two repeated three four", "one point two recurring three four", "one point two repetend three four" or "one point two into infinity three four". Likewise, 11.1886792452830 may be read "eleven point repeating one double eight six seven nine two four five two eight three zero", "eleven point repeated one double eight six seven nine two four five two eight three zero", "eleven point recurring one double eight six seven nine two four five two eight three zero" "eleven point repetend one double eight six seven nine two four five two eight three zero" or "eleven point into infinity one double eight six seven nine two four five two eight three zero". <div class="mw-heading mw-heading3"><h3 id="Decimal_expansion_and_recurrence_sequence">Decimal expansion and recurrence sequence</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=3" title="Edit section: Decimal expansion and recurrence sequence">edit</a>]</div> In order to convert a <a href="/wiki/Rational_number" title="Rational number">rational number</a> represented as a fraction into decimal form, one may use <a href="/wiki/Long_division" title="Long division">long division</a>. For example, consider the rational number <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/74⁠: <pre> 0.0675 74 ) 5.00000 4.44 560 518 420 370 500 </pre> etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore, the decimal repeats: 0.0675675675.... For any integer fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠A/B⁠, the remainder at step k, for any positive integer k, is A × 10k (modulo B). <div class="mw-heading mw-heading3"><h3 id="Every_rational_number_is_either_a_terminating_or_repeating_decimal">Every rational number is either a terminating or repeating decimal</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=4" title="Edit section: Every rational number is either a terminating or repeating decimal">edit</a>]</div> For any given divisor, only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73. If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called "period", is defined to be 0. If 0 never occurs as a remainder, then the division process continues forever, and eventually, a remainder must occur that has occurred before. The next step in the division will yield the same new digit in the quotient, and the same new remainder, as the previous time the remainder was the same. Therefore, the following division will repeat the same results. The repeating sequence of digits is called "repetend" which has a certain length greater than 0, also called "period".<a href="#cite_note-5">[5]</a> In base 10, a fraction has a repeating decimal if and only if <a href="/wiki/In_lowest_terms" class="mw-redirect" title="In lowest terms">in lowest terms</a>, its denominator has any prime factors besides 2 or 5, or in other words, cannot be expressed as 2m 5n, where m and n are non-negative integers. <div class="mw-heading mw-heading3"><h3 id="Every_repeating_or_terminating_decimal_is_a_rational_number">Every repeating or terminating decimal is a rational number</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=5" title="Edit section: Every repeating or terminating decimal is a rational number">edit</a>]</div> Each repeating decimal number satisfies a <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a> with integer coefficients, and its unique solution is a rational number. In the example above, α = 5.8144144144... satisfies the equation <dl><dd><table> <tbody><tr> <td nowrap="">10000α − 10α </td> <td nowrap="">= 58144.144144... − 58.144144... </td></tr> <tr> <td align="right">9990α</td> <td>= 58086 </td></tr> <tr> <td align="right">Therefore, α</td> <td>= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠58086/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3227/555⁠ </td></tr></tbody></table></dd></dl> The process of how to find these integer coefficients is described <a href="#Converting_repeating_decimals_to_fractions">below</a>. <div class="mw-heading mw-heading4"><h4 id="Formal_proof">Formal proof</h4>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=6" title="Edit section: Formal proof">edit</a>]</div> Given a repeating decimal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a.b{\overline {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>.</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a.b{\overline {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a54142b38eef55044f1aa4adc502ee0a13a066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.811ex; height:2.343ex;" alt="{\displaystyle x=a.b{\overline {c}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> are groups of digits, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\lceil {\log _{10}b}\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo fence="false" stretchy="false">⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> </mrow> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\lceil {\log _{10}b}\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdf45b77c6dfd8c6759cff84d486cf27849b14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.791ex; height:2.843ex;" alt="{\displaystyle n=\lceil {\log _{10}b}\rceil }">, the number of digits of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. Multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeff06a7c9ad9455cb809047cfc97a92c51e1bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle 10^{n}}"> separates the repeating and terminating groups: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{n}x=ab.{\bar {c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{n}x=ab.{\bar {c}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f7466a59e4142a7edc3df474358a079229996a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.171ex; height:2.343ex;" alt="{\displaystyle 10^{n}x=ab.{\bar {c}}.}"> If the decimals terminate (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ee918699d0cb4b8c633cc1f520a8a7a174f44a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=0}">), the proof is complete.<a href="#cite_note-6">[6]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be27396bd0e62003728d08329a8767eee94409e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.676ex;" alt="{\displaystyle c\neq 0}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"> digits, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y.{\bar {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y.{\bar {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86fd9b77eaca7c16ffccee921995ccbede1a858c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.909ex; height:2.343ex;" alt="{\displaystyle x=y.{\bar {c}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e6f12a950fae74d6a37b86f7a4bca9174475e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {Z} }"> is a terminating group of digits. Then, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=d_{1}d_{2}\,...d_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=d_{1}d_{2}\,...d_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c2f7f6257c9705568373b86aa139740d56925d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.418ex; height:2.509ex;" alt="{\displaystyle c=d_{1}d_{2}\,...d_{k}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"> denotes the i-th digit, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y+\sum _{n=1}^{\infty }{\frac {c}{{(10^{k})}^{n}}}=y+\left(c\sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}\right)-c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y+\sum _{n=1}^{\infty }{\frac {c}{{(10^{k})}^{n}}}=y+\left(c\sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}\right)-c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1500a68107709fde5a6e2e4956debe04b11f3d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.127ex; height:7.509ex;" alt="{\displaystyle x=y+\sum _{n=1}^{\infty }{\frac {c}{{(10^{k})}^{n}}}=y+\left(c\sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}\right)-c.}"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}={\frac {1}{1-10^{-k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}={\frac {1}{1-10^{-k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c68a029f27a5a9b3f4d516712d861766c9d80e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.32ex; height:4.509ex;" alt="{\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}={\frac {1}{1-10^{-k}}}}">,<a href="#cite_note-7">[7]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y-c+{\frac {10^{k}c}{10^{k}-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>−</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>c</mi> </mrow> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y-c+{\frac {10^{k}c}{10^{k}-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae520e5a03c6276f28b18dc2984348d4f31cb69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.171ex; height:6.343ex;" alt="{\displaystyle x=y-c+{\frac {10^{k}c}{10^{k}-1}}.}"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is the sum of an integer (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31b8fa48b1c33a27478e0118eacc4c418169c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.003ex; height:2.343ex;" alt="{\displaystyle y-c}">) and a rational number (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {10^{k}c}{10^{k}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>c</mi> </mrow> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {10^{k}c}{10^{k}-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768bdcdf2721d71820459b7f5f35c74b4687f538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.44ex; height:4.676ex;" alt="{\textstyle {\frac {10^{k}c}{10^{k}-1}}}">), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is also rational.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading2"><h2 id="Table_of_values">Table of values</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=7" title="Edit section: Table of values">edit</a>]</div> <div><ul> <li style="display: inline-table;"> <table class="wikitable"> <tbody><tr> <th class="nowrap ts-vertical-header is-valign-bot" style=""><div style=""><style data-mw-deduplicate="TemplateStyles:r1221560606">@supports(writing-mode:vertical-rl){.mw-parser-output .ts-vertical-header{line-height:1;max-width:1em;padding:0.4em;vertical-align:bottom;width:1em}html.client-js .mw-parser-output .sortable:not(.jquery-tablesorter) .ts-vertical-header:not(.unsortable),html.client-js .mw-parser-output .ts-vertical-header.headerSort{background-position:50%.4em;padding-right:0.4em;padding-top:21px}.mw-parser-output .ts-vertical-header.is-valign-top{vertical-align:top}.mw-parser-output .ts-vertical-header.is-valign-middle{vertical-align:middle}.mw-parser-output .ts-vertical-header.is-normal{font-weight:normal}.mw-parser-output .ts-vertical-header>*{display:inline-block;transform:rotate(180deg);writing-mode:vertical-rl}@supports(writing-mode:sideways-lr){.mw-parser-output .ts-vertical-header>*{transform:none;writing-mode:sideways-lr}}}</style>fraction</div> </th> <th>decimal expansion </th> <th><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style>ℓ10 </th> <th>binary expansion </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488">ℓ2 </th></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ </td> <td>0.5 </td> <td>0 </td> <td>0.1 </td> <td>0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠ </td> <td>0.3 </td> <td>1 </td> <td>0.01 </td> <td>2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ </td> <td>0.25 </td> <td>0 </td> <td>0.01 </td> <td>0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/5⁠ </td> <td>0.2 </td> <td>0 </td> <td>0.0011 </td> <td>4 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ </td> <td>0.16 </td> <td>1 </td> <td>0.001 </td> <td>2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ </td> <td>0.142857 </td> <td>6 </td> <td>0.001 </td> <td>3 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠ </td> <td>0.125 </td> <td>0 </td> <td>0.001 </td> <td>0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/9⁠ </td> <td>0.1 </td> <td>1 </td> <td>0.000111 </td> <td>6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/10⁠ </td> <td>0.1 </td> <td>0 </td> <td>0.00011 </td> <td>4 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/11⁠ </td> <td>0.09 </td> <td>2 </td> <td>0.0001011101 </td> <td>10 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/12⁠ </td> <td>0.083 </td> <td>1 </td> <td>0.0001 </td> <td>2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/13⁠ </td> <td>0.076923 </td> <td>6 </td> <td>0.000100111011 </td> <td>12 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/14⁠ </td> <td>0.0714285 </td> <td>6 </td> <td>0.0001 </td> <td>3 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/15⁠ </td> <td>0.06 </td> <td>1 </td> <td>0.0001 </td> <td>4 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/16⁠ </td> <td>0.0625 </td> <td>0 </td> <td>0.0001 </td> <td>0 </td></tr></tbody></table> </li> <li style="display: inline-table;"> <table class="wikitable"> <tbody><tr> <th class="nowrap ts-vertical-header is-valign-bot" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606">fraction</div> </th> <th>decimal expansion </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488">ℓ10 </th></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/17⁠ </td> <td>0.0588235294117647 </td> <td style="text-align:right">16 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/18⁠ </td> <td>0.05 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/19⁠ </td> <td>0.052631578947368421 </td> <td style="text-align:right">18 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/20⁠ </td> <td>0.05 </td> <td style="text-align:right">0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/21⁠ </td> <td>0.047619 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/22⁠ </td> <td>0.045 </td> <td style="text-align:right">2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/23⁠ </td> <td>0.0434782608695652173913 </td> <td style="text-align:right">22 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/24⁠ </td> <td>0.0416 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/25⁠ </td> <td>0.04 </td> <td style="text-align:right">0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/26⁠ </td> <td>0.0384615 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/27⁠ </td> <td>0.037 </td> <td style="text-align:right">3 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/28⁠ </td> <td>0.03571428 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/29⁠ </td> <td>0.0344827586206896551724137931 </td> <td style="text-align:right">28 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠ </td> <td>0.03 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/31⁠ </td> <td>0.032258064516129 </td> <td style="text-align:right">15 </td></tr></tbody></table> </li> <li style="display: inline-table;"> <table class="wikitable"> <tbody><tr> <th class="nowrap ts-vertical-header is-valign-bot" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606">fraction</div> </th> <th>decimal expansion </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488">ℓ10 </th></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/32⁠ </td> <td>0.03125 </td> <td style="text-align:right">0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/33⁠ </td> <td>0.03 </td> <td style="text-align:right">2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/34⁠ </td> <td>0.02941176470588235 </td> <td style="text-align:right">16 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/35⁠ </td> <td>0.0285714 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/36⁠ </td> <td>0.027 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/37⁠ </td> <td>0.027 </td> <td style="text-align:right">3 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/38⁠ </td> <td>0.0263157894736842105 </td> <td style="text-align:right">18 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/39⁠ </td> <td>0.025641 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/40⁠ </td> <td>0.025 </td> <td style="text-align:right">0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/41⁠ </td> <td>0.02439 </td> <td style="text-align:right">5 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/42⁠ </td> <td>0.0238095 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/43⁠ </td> <td>0.023255813953488372093 </td> <td style="text-align:right">21 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/44⁠ </td> <td>0.0227 </td> <td style="text-align:right">2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/45⁠ </td> <td>0.02 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/46⁠ </td> <td>0.02173913043478260869565 </td> <td style="text-align:right">22 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/47⁠ </td> <td>0.0212765957446808510638297872340425531914893617 </td> <td style="text-align:right">46 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/48⁠ </td> <td>0.02083 </td> <td style="text-align:right">1 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/49⁠ </td> <td>0.020408163265306122448979591836734693877551 </td> <td style="text-align:right">42 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/50⁠ </td> <td>0.02 </td> <td style="text-align:right">0 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/51⁠ </td> <td>0.0196078431372549 </td> <td style="text-align:right">16 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/52⁠ </td> <td>0.01923076 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/53⁠ </td> <td>0.0188679245283 </td> <td style="text-align:right">13 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/54⁠ </td> <td>0.0185 </td> <td style="text-align:right">3 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/55⁠ </td> <td>0.018 </td> <td style="text-align:right">2 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/56⁠ </td> <td>0.017857142 </td> <td style="text-align:right">6 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/57⁠ </td> <td>0.017543859649122807 </td> <td style="text-align:right">18 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/58⁠ </td> <td>0.01724137931034482758620689655 </td> <td style="text-align:right">28 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/59⁠ </td> <td>0.0169491525423728813559322033898305084745762711864406779661 </td> <td style="text-align:right">58 </td></tr> <tr> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/60⁠ </td> <td>0.016 </td> <td style="text-align:right">1 </td></tr></tbody></table> </li> </ul></div> Thereby fraction is the <a href="/wiki/Unit_fraction" title="Unit fraction">unit fraction</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠ and ℓ10 is the length of the (decimal) repetend. The lengths ℓ10(n) of the decimal repetends of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠, n = 1, 2, 3, ..., are: <dl><dd>0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1, 6, 22, 15, 46, 18, 1, 96, 42, 2, 0... (sequence <a href="//oeis.org/A051626" class="extiw" title="oeis:A051626">A051626</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> For comparison, the lengths ℓ2(n) of the <a href="/wiki/Binary_number#Representation" title="Binary number">binary</a> repetends of the fractions <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠, n = 1, 2, 3, ..., are: <dl><dd>0, 0, 2, 0, 4, 2, 3, 0, 6, 4, 10, 2, 12, 3, 4, 0, 8, 6, 18, 4, 6, 10, 11, 2, 20, 12, 18, 3, 28, 4, 5, 0, 10, 8, 12, 6, 36, 18, 12, 4, 20, 6, 14, 10, 12, 11, ... (=<a href="//oeis.org/A007733" class="extiw" title="oeis:A007733">A007733</a>[n], if n not a power of 2 else =0).</dd></dl> The decimal repetends of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠, n = 1, 2, 3, ..., are: <dl><dd>0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 09, 3, 076923, 714285, 6, 0, 0588235294117647, 5, 052631578947368421, 0, 047619, 45, 0434782608695652173913, 6, 0, 384615, 037, 571428, 0344827586206896551724137931, 3, 032258064516129, 0, 03, 2941176470588235, 285714... (sequence <a href="//oeis.org/A036275" class="extiw" title="oeis:A036275">A036275</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> The decimal repetend lengths of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠, p = 2, 3, 5, ... (nth prime), are: <dl><dd>0, 1, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79... (sequence <a href="//oeis.org/A002371" class="extiw" title="oeis:A002371">A002371</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> The least primes p for which <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ has decimal repetend length n, n = 1, 2, 3, ..., are: <dl><dd>3, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 99990001, 21401, 859, 757, 29, 3191, 211, 2791, 353, 67, 103, 71, 999999000001, 2028119, 909090909090909091, 900900900900990990990991, 1676321, 83, 127, 173... (sequence <a href="//oeis.org/A007138" class="extiw" title="oeis:A007138">A007138</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> The least primes p for which <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠k/p⁠ has n different cycles (1 ≤ k ≤ p−1), n = 1, 2, 3, ..., are: <dl><dd>7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931... (sequence <a href="//oeis.org/A054471" class="extiw" title="oeis:A054471">A054471</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Fractions_with_prime_denominators">Fractions with prime denominators</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=8" title="Edit section: Fractions with prime denominators">edit</a>]</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/Reciprocals_of_primes" title="Reciprocals of primes">Reciprocals of primes</a></div> A fraction <a href="/wiki/In_lowest_terms" class="mw-redirect" title="In lowest terms">in lowest terms</a> with a <a href="/wiki/Prime_number" title="Prime number">prime</a> denominator other than 2 or 5 (i.e. <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal segment) of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ is equal to the <a href="/wiki/Multiplicative_order" title="Multiplicative order">order</a> of 10 modulo p. If 10 is a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive root</a> modulo p, then the repetend length is equal to p − 1; if not, then the repetend length is a factor of p − 1. This result can be deduced from <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>, which states that 10p−1 ≡ 1 (mod p). The base-10 <a href="/wiki/Digital_root" title="Digital root">digital root</a> of the repetend of the reciprocal of any prime number greater than 5 is 9.<a href="#cite_note-9">[9]</a> If the repetend length of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ for prime p is equal to p − 1 then the repetend, expressed as an integer, is called a cyclic number. <div class="mw-heading mw-heading3"><h3 id="Cyclic_numbers">Cyclic numbers</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=9" title="Edit section: Cyclic numbers">edit</a>]</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/Cyclic_number" title="Cyclic number">Cyclic number</a></div> Examples of fractions belonging to this group are: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ = 0.142857, 6 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/17⁠ = 0.0588235294117647, 16 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/19⁠ = 0.052631578947368421, 18 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/23⁠ = 0.0434782608695652173913, 22 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/29⁠ = 0.0344827586206896551724137931, 28 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/47⁠ = 0.0212765957446808510638297872340425531914893617, 46 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/59⁠ = 0.0169491525423728813559322033898305084745762711864406779661, 58 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/61⁠ = 0.016393442622950819672131147540983606557377049180327868852459, 60 repeating digits</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/97⁠ = 0.010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567, 96 repeating digits</li></ul> The list can go on to include the fractions <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/109⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/113⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/131⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/149⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/167⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/179⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/181⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/193⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/223⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/229⁠, etc. (sequence <a href="//oeis.org/A001913" class="extiw" title="oeis:A001913">A001913</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). Every proper multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ = 1 × 0.142857 = 0.142857</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/7⁠ = 2 × 0.142857 = 0.285714</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/7⁠ = 3 × 0.142857 = 0.428571</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/7⁠ = 4 × 0.142857 = 0.571428</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/7⁠ = 5 × 0.142857 = 0.714285</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/7⁠ = 6 × 0.142857 = 0.857142</li></ul> The reason for the cyclic behavior is apparent from an arithmetic exercise of long division of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠: the sequential remainders are the cyclic sequence {1, 3, 2, 6, 4, 5}. See also the article <a href="/wiki/142,857" class="mw-redirect" title="142,857">142,857</a> for more properties of this cyclic number. A fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in <a href="/wiki/Nines%27_complement" class="mw-redirect" title="Nines' complement">nines' complement</a> form. For example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ starts '142' and is followed by '857' while <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/7⁠ (by rotation) starts '857' followed by its nines' complement '142'. The rotation of the repetend of a cyclic number always happens in such a way that each successive repetend is a bigger number than the previous one. In the succession above, for instance, we see that 0.142857... < 0.285714... < 0.428571... < 0.571428... < 0.714285... < 0.857142.... This, for cyclic fractions with long repetends, allows us to easily predict what the result of multiplying the fraction by any natural number n will be, as long as the repetend is known. A proper prime is a prime p which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repetend with length p − 1. In such primes, each digit 0, 1,..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p − 1/10⁠ times). They are:<a href="#cite_note-10">[10]</a>: 166  <dl><dd>61, 131, 181, 461, 491, 541, 571, 701, 811, 821, 941, 971, 1021, 1051, 1091, 1171, 1181, 1291, 1301, 1349, 1381, 1531, 1571, 1621, 1741, 1811, 1829, 1861,... (sequence <a href="//oeis.org/A073761" class="extiw" title="oeis:A073761">A073761</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> A prime is a proper prime if and only if it is a <a href="/wiki/Full_reptend_prime" title="Full reptend prime">full reptend prime</a> and <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">congruent</a> to 1 mod 10. If a prime p is both <a href="/wiki/Full_reptend_prime" title="Full reptend prime">full reptend prime</a> and <a href="/wiki/Safe_prime" class="mw-redirect" title="Safe prime">safe prime</a>, then <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ will produce a stream of p − 1 <a href="/wiki/Pseudo-random_numbers" class="mw-redirect" title="Pseudo-random numbers">pseudo-random digits</a>. Those primes are <dl><dd>7, 23, 47, 59, 167, 179, 263, 383, 503, 863, 887, 983, 1019, 1367, 1487, 1619, 1823, 2063... (sequence <a href="//oeis.org/A000353" class="extiw" title="oeis:A000353">A000353</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Other_reciprocals_of_primes">Other reciprocals of primes</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=10" title="Edit section: Other reciprocals of primes">edit</a>]</div> Some reciprocals of primes that do not generate cyclic numbers are: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠ = 0.3, which has a period (repetend length) of 1.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/11⁠ = 0.09, which has a period of two.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/13⁠ = 0.076923, which has a period of six.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/31⁠ = 0.032258064516129, which has a period of 15.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/37⁠ = 0.027, which has a period of three.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/41⁠ = 0.02439, which has a period of five.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/43⁠ = 0.023255813953488372093, which has a period of 21.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/53⁠ = 0.0188679245283, which has a period of 13.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/67⁠ = 0.014925373134328358208955223880597, which has a period of 33.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/71⁠ = 0.01408450704225352112676058338028169, which has a period of 35.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/73⁠ = 0.01369863, which has a period of eight.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/79⁠ = 0.0126582278481, which has a period of 13.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/83⁠ = 0.01204819277108433734939759036144578313253, which has a period of 41.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/89⁠ = 0.01123595505617977528089887640449438202247191, which has a period of 44.</li></ul> (sequence <a href="//oeis.org/A006559" class="extiw" title="oeis:A006559">A006559</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) The reason is that 3 is a divisor of 9, 11 is a divisor of 99, 41 is a divisor of 99999, etc. To find the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠, we can check whether the prime p divides some number 999...999 in which the number of digits divides p − 1. Since the period is never greater than p − 1, we can obtain this by calculating <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠10p−1 − 1/p⁠. For example, for 11 we get <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {10^{11-1}-1}{11}}=909090909}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mn>11</mn> </mfrac> </mrow> <mo>=</mo> <mn>909090909</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {10^{11-1}-1}{11}}=909090909}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ba021fc547132937a1fa2b7e586ba6830b9e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.701ex; height:5.676ex;" alt="{\displaystyle {\frac {10^{11-1}-1}{11}}=909090909}"></dd></dl> and then by inspection find the repetend 09 and period of 2. Those reciprocals of primes can be associated with several sequences of repeating decimals. For example, the multiples of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/13⁠ can be divided into two sets, with different repetends. The first set is: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/13⁠ = 0.076923</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠10/13⁠ = 0.769230</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/13⁠ = 0.692307</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12/13⁠ = 0.923076</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/13⁠ = 0.230769</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/13⁠ = 0.307692</li></ul> where the repetend of each fraction is a cyclic re-arrangement of 076923. The second set is: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/13⁠ = 0.153846</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/13⁠ = 0.538461</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/13⁠ = 0.384615</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠11/13⁠ = 0.846153</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/13⁠ = 0.461538</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠8/13⁠ = 0.615384</li></ul> where the repetend of each fraction is a cyclic re-arrangement of 153846. In general, the set of proper multiples of reciprocals of a prime p consists of n subsets, each with repetend length k, where nk = p − 1. <div class="mw-heading mw-heading3"><h3 id="Totient_rule">Totient rule</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=11" title="Edit section: Totient rule">edit</a>]</div> For an arbitrary integer n, the length L(n) of the decimal repetend of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n⁠ divides φ(n), where φ is the <a href="/wiki/Totient_function" class="mw-redirect" title="Totient function">totient function</a>. The length is equal to φ(n) if and only if 10 is a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive root modulo n</a>.<a href="#cite_note-11">[11]</a> In particular, it follows that L(p) = p − 1 <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> p is a prime and 10 is a primitive root modulo p. Then, the decimal expansions of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/p⁠ for n = 1, 2, ..., p − 1, all have period p − 1 and differ only by a cyclic permutation. Such numbers p are called <a href="/wiki/Full_repetend_prime" class="mw-redirect" title="Full repetend prime">full repetend primes</a>. <div class="mw-heading mw-heading2"><h2 id="Reciprocals_of_composite_integers_coprime_to_10">Reciprocals of composite integers coprime to 10</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=12" title="Edit section: Reciprocals of composite integers coprime to 10">edit</a>]</div> If p is a prime other than 2 or 5, the decimal representation of the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p2⁠ repeats: <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/49⁠ = 0.020408163265306122448979591836734693877551.</dd></dl> The period (repetend length) L(49) must be a factor of λ(49) = 42, where λ(n) is known as the <a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael function</a>. This follows from <a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael's theorem</a> which states that if n is a positive integer then λ(n) is the smallest integer m such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{m}\equiv 1{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}\equiv 1{\pmod {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6295499efc8f39a0cd7ec690788edcb794eb2bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.245ex; height:2.843ex;" alt="{\displaystyle a^{m}\equiv 1{\pmod {n}}}"></dd></dl> for every integer a that is <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to n. The period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p2⁠ is usually pTp, where Tp is the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠. There are three known primes for which this is not true, and for those the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p2⁠ is the same as the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ because p2 divides 10p−1−1. These three primes are 3, 487, and 56598313 (sequence <a href="//oeis.org/A045616" class="extiw" title="oeis:A045616">A045616</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).<a href="#cite_note-12">[12]</a> Similarly, the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pk⁠ is usually pk–1Tp If p and q are primes other than 2 or 5, the decimal representation of the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pq⁠ repeats. An example is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/119⁠: <dl><dd>119 = 7 × 17</dd> <dd>λ(7 × 17) = <a href="/wiki/Least_common_multiple" title="Least common multiple">LCM</a>(λ(7), λ(17)) = LCM(6, 16) = 48,</dd></dl> where LCM denotes the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a>. The period T of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pq⁠ is a factor of λ(pq) and it happens to be 48 in this case: <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/119⁠ = 0.008403361344537815126050420168067226890756302521.</dd></dl> The period T of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pq⁠ is LCM(Tp, Tq), where Tp is the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ and Tq is the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/q⁠. If p, q, r, etc. are primes other than 2 or 5, and k, ℓ, m, etc. are positive integers, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{p^{k}q^{\ell }r^{m}\cdots }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{p^{k}q^{\ell }r^{m}\cdots }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8547e2bb40bd8bffec2fba6a9f5258738f1515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.926ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{p^{k}q^{\ell }r^{m}\cdots }}}"></dd></dl> is a repeating decimal with a period of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LCM} (T_{p^{k}},T_{q^{\ell }},T_{r^{m}},\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>LCM</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {LCM} (T_{p^{k}},T_{q^{\ell }},T_{r^{m}},\ldots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98204bc07ab0c0696c6fb10ab8eb9e4d79df273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.913ex; height:3.176ex;" alt="{\displaystyle \operatorname {LCM} (T_{p^{k}},T_{q^{\ell }},T_{r^{m}},\ldots )}"></dd></dl> where Tpk, Tqℓ, Trm,... are respectively the period of the repeating decimals <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pk⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/qℓ⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/rm⁠,... as defined above. <div class="mw-heading mw-heading2"><h2 id="Reciprocals_of_integers_not_coprime_to_10">Reciprocals of integers not coprime to 10</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=13" title="Edit section: Reciprocals of integers not coprime to 10">edit</a>]</div> An integer that is not coprime to 10 but has a prime factor other than 2 or 5 has a reciprocal that is eventually periodic, but with a non-repeating sequence of digits that precede the repeating part. The reciprocal can be expressed as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2^{a}\cdot 5^{b}p^{k}q^{\ell }\cdots }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2^{a}\cdot 5^{b}p^{k}q^{\ell }\cdots }}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad675249daaa2062aeef4b8bfc605e1eab48c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.279ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2^{a}\cdot 5^{b}p^{k}q^{\ell }\cdots }}\,,}"></dd></dl> where a and b are not both zero. This fraction can also be expressed as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5^{a-b}}{10^{a}p^{k}q^{\ell }\cdots }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> </msup> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5^{a-b}}{10^{a}p^{k}q^{\ell }\cdots }}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16feb9226afa4118238e99118b7f17f43a4e156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.663ex; height:6.343ex;" alt="{\displaystyle {\frac {5^{a-b}}{10^{a}p^{k}q^{\ell }\cdots }}\,,}"></dd></dl> if a > b, or as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2^{b-a}}{10^{b}p^{k}q^{\ell }\cdots }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </msup> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2^{b-a}}{10^{b}p^{k}q^{\ell }\cdots }}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb274c50fb84c523ccfbf1a569e4b2da16fcd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.498ex; height:6.676ex;" alt="{\displaystyle {\frac {2^{b-a}}{10^{b}p^{k}q^{\ell }\cdots }}\,,}"></dd></dl> if b > a, or as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{10^{a}p^{k}q^{\ell }\cdots }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10^{a}p^{k}q^{\ell }\cdots }}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8d4176cddb1b5271eb8be70f959a83df967793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.663ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{10^{a}p^{k}q^{\ell }\cdots }}\,,}"></dd></dl> if a = b. The decimal has: <ul><li>An initial transient of max(a, b) digits after the decimal point. Some or all of the digits in the transient can be zeros.</li> <li>A subsequent repetend which is the same as that for the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/pk qℓ ⋯⁠.</li></ul> For example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/28⁠ = 0.03571428: <ul><li>a = 2, b = 0, and the other factors pk qℓ ⋯ = 7</li> <li>there are 2 initial non-repeating digits, 03; and</li> <li>there are 6 repeating digits, 571428, the same amount as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ has.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Converting_repeating_decimals_to_fractions">Converting repeating decimals to fractions</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=14" title="Edit section: Converting repeating decimals to fractions">edit</a>]</div> Given a repeating decimal, it is possible to calculate the fraction that produces it. For example: <dl><dd><table> <tbody><tr> <td style="text-align:right;width:3em"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></td> <td style="width:12em"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =0.333333\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>0.333333</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =0.333333\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdd989bed8804e4f3fc1df7afd8ac1ec25c5ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.348ex; height:2.176ex;" alt="{\displaystyle =0.333333\ldots }"> </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d55a1c1904762c24cf43f5000da77696052001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle 10x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =3.333333\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>3.333333</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =3.333333\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f399f536ec7a478ddb0a5fe75d0de4d4eba6b504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.348ex; height:2.176ex;" alt="{\displaystyle =3.333333\ldots }"></td> <td>(multiply each side of the above line by 10) </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a157790e3ea2ba51d6ae2143969bbdab80f68d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 9x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40a8447b4be595459a57d399a1e490c08c1113e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =3}"></td> <td>(subtract the 1st line from the 2nd) </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {3}{9}}={\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {3}{9}}={\frac {1}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f166bb95037311c490966543d221dca7ab0a0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.549ex; height:5.176ex;" alt="{\displaystyle ={\frac {3}{9}}={\frac {1}{3}}}"></td> <td>(reduce to lowest terms) </td></tr></tbody></table></dd></dl> Another example: <dl><dd><table> <tbody><tr> <td style="text-align:right;width:3em"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></td> <td style="width:12em"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\ \ \ \ 0.836363636\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mn>0.836363636</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\ \ \ \ 0.836363636\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39be152f9af3cdb84932f53ba6d91baa0424d1fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.158ex; height:2.176ex;" alt="{\displaystyle =\ \ \ \ 0.836363636\ldots }"> </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d55a1c1904762c24cf43f5000da77696052001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.176ex;" alt="{\displaystyle 10x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\ \ \ \ 8.36363636\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mn>8.36363636</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\ \ \ \ 8.36363636\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29784217f7e5fb5b5ef0f1c05fac012c96b4928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.995ex; height:2.176ex;" alt="{\displaystyle =\ \ \ \ 8.36363636\ldots }"></td> <td>(move decimal to start of repetition = move by 1 place = multiply by 10) </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1000x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1000</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1000x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e86ef591fc77aecd41122f45f3752137ba31631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.98ex; height:2.176ex;" alt="{\displaystyle 1000x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =836.36363636\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>836.36363636</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =836.36363636\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5676af02ed9aa90c2a526acd3b1625a87c1306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.997ex; height:2.176ex;" alt="{\displaystyle =836.36363636\ldots }"></td> <td>(collate 2nd repetition here with 1st above = move by 2 places = multiply by 100) </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 990x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>990</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 990x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f255a0f25922edbc825f8076f484764f879ea17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.817ex; height:2.176ex;" alt="{\displaystyle 990x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =828}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>828</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =828}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550332868c429a966447487043afa51a5b645614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle =828}"></td> <td>(subtract to clear decimals) </td></tr> <tr> <td style="text-align:right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {828}{990}}={\frac {18\cdot 46}{18\cdot 55}}={\frac {46}{55}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>828</mn> <mn>990</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>18</mn> <mo>⋅</mo> <mn>46</mn> </mrow> <mrow> <mn>18</mn> <mo>⋅</mo> <mn>55</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>46</mn> <mn>55</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {828}{990}}={\frac {18\cdot 46}{18\cdot 55}}={\frac {46}{55}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4eabe6884faafa39c1fa41c340b1ece5f8d6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.3ex; height:5.176ex;" alt="{\displaystyle ={\frac {828}{990}}={\frac {18\cdot 46}{18\cdot 55}}={\frac {46}{55}}}"></td> <td>(reduce to lowest terms) </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="A_shortcut">A shortcut</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=15" title="Edit section: A shortcut">edit</a>]</div> The procedure below can be applied in particular if the repetend has n digits, all of which are 0 except the final one which is 1. For instance for n = 7: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=0.000000100000010000001\ldots \\10^{7}x&=1.000000100000010000001\ldots \\\left(10^{7}-1\right)x=9999999x&=1\\x&={\frac {1}{10^{7}-1}}={\frac {1}{9999999}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.000000100000010000001</mn> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1.000000100000010000001</mn> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>=</mo> <mn>9999999</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9999999</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=0.000000100000010000001\ldots \\10^{7}x&=1.000000100000010000001\ldots \\\left(10^{7}-1\right)x=9999999x&=1\\x&={\frac {1}{10^{7}-1}}={\frac {1}{9999999}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ef6b650d7e3461209779df0ecaf648d05515e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.029ex; margin-bottom: -0.309ex; width:56.975ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}x&=0.000000100000010000001\ldots \\10^{7}x&=1.000000100000010000001\ldots \\\left(10^{7}-1\right)x=9999999x&=1\\x&={\frac {1}{10^{7}-1}}={\frac {1}{9999999}}\end{aligned}}}"></dd></dl> So this particular repeating decimal corresponds to the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/10n − 1⁠, where the denominator is the number written as n 9s. Knowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation. For example, one could reason: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}7.48181818\ldots &=7.3+0.18181818\ldots \\[8pt]&={\frac {73}{10}}+{\frac {18}{99}}={\frac {73}{10}}+{\frac {9\cdot 2}{9\cdot 11}}={\frac {73}{10}}+{\frac {2}{11}}\\[12pt]&={\frac {11\cdot 73+10\cdot 2}{10\cdot 11}}={\frac {823}{110}}\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="1.1em 1.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>7.48181818</mn> <mo>…</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>7.3</mn> <mo>+</mo> <mn>0.18181818</mn> <mo>…</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>73</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>99</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>73</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>9</mn> <mo>⋅</mo> <mn>11</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>73</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>11</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>⋅</mo> <mn>73</mn> <mo>+</mo> <mn>10</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>10</mn> <mo>⋅</mo> <mn>11</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>823</mn> <mn>110</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}7.48181818\ldots &=7.3+0.18181818\ldots \\[8pt]&={\frac {73}{10}}+{\frac {18}{99}}={\frac {73}{10}}+{\frac {9\cdot 2}{9\cdot 11}}={\frac {73}{10}}+{\frac {2}{11}}\\[12pt]&={\frac {11\cdot 73+10\cdot 2}{10\cdot 11}}={\frac {823}{110}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cf612540b8255ca0ae162cc6f8ccad16a2ed93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:54.595ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}7.48181818\ldots &=7.3+0.18181818\ldots \\[8pt]&={\frac {73}{10}}+{\frac {18}{99}}={\frac {73}{10}}+{\frac {9\cdot 2}{9\cdot 11}}={\frac {73}{10}}+{\frac {2}{11}}\\[12pt]&={\frac {11\cdot 73+10\cdot 2}{10\cdot 11}}={\frac {823}{110}}\end{aligned}}}"></dd></dl> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}11.18867924528301886792452830\ldots &=11+0.18867924528301886792452830\ldots \\[8pt]&=11+{\frac {10}{53}}={\frac {11\cdot 53+10}{53}}={\frac {593}{53}}\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="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>11.18867924528301886792452830</mn> <mo>…</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>11</mn> <mo>+</mo> <mn>0.18867924528301886792452830</mn> <mo>…</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>11</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>53</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>⋅</mo> <mn>53</mn> <mo>+</mo> <mn>10</mn> </mrow> <mn>53</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>593</mn> <mn>53</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}11.18867924528301886792452830\ldots &=11+0.18867924528301886792452830\ldots \\[8pt]&=11+{\frac {10}{53}}={\frac {11\cdot 53+10}{53}}={\frac {593}{53}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebeebd569c5d473703b099d8068247a0def48d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:80.465ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}11.18867924528301886792452830\ldots &=11+0.18867924528301886792452830\ldots \\[8pt]&=11+{\frac {10}{53}}={\frac {11\cdot 53+10}{53}}={\frac {593}{53}}\end{aligned}}}"></dd></dl> It is possible to get a general formula expressing a repeating decimal with an n-digit period (repetend length), beginning right after the decimal point, as a fraction: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=0.{\overline {a_{1}a_{2}\cdots a_{n}}}\\10^{n}x&=a_{1}a_{2}\cdots a_{n}.{\overline {a_{1}a_{2}\cdots a_{n}}}\\[5pt]\left(10^{n}-1\right)x=99\cdots 99x&=a_{1}a_{2}\cdots a_{n}\\[5pt]x&={\frac {a_{1}a_{2}\cdots a_{n}}{10^{n}-1}}={\frac {a_{1}a_{2}\cdots a_{n}}{99\cdots 99}}\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 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>=</mo> <mn>99</mn> <mo>⋯</mo> <mn>99</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mn>99</mn> <mo>⋯</mo> <mn>99</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=0.{\overline {a_{1}a_{2}\cdots a_{n}}}\\10^{n}x&=a_{1}a_{2}\cdots a_{n}.{\overline {a_{1}a_{2}\cdots a_{n}}}\\[5pt]\left(10^{n}-1\right)x=99\cdots 99x&=a_{1}a_{2}\cdots a_{n}\\[5pt]x&={\frac {a_{1}a_{2}\cdots a_{n}}{10^{n}-1}}={\frac {a_{1}a_{2}\cdots a_{n}}{99\cdots 99}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2515e466c7582dd33e653060a6efdebcff65665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:53.296ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}x&=0.{\overline {a_{1}a_{2}\cdots a_{n}}}\\10^{n}x&=a_{1}a_{2}\cdots a_{n}.{\overline {a_{1}a_{2}\cdots a_{n}}}\\[5pt]\left(10^{n}-1\right)x=99\cdots 99x&=a_{1}a_{2}\cdots a_{n}\\[5pt]x&={\frac {a_{1}a_{2}\cdots a_{n}}{10^{n}-1}}={\frac {a_{1}a_{2}\cdots a_{n}}{99\cdots 99}}\end{aligned}}}"></dd></dl> More explicitly, one gets the following cases: If the repeating decimal is between 0 and 1, and the repeating block is n digits long, first occurring right after the decimal point, then the fraction (not necessarily reduced) will be the integer number represented by the n-digit block divided by the one represented by n 9s. For example, <ul><li>0.444444... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/9⁠ since the repeating block is 4 (a 1-digit block),</li> <li>0.565656... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠56/99⁠ since the repeating block is 56 (a 2-digit block),</li> <li>0.012012... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12/999⁠ since the repeating block is 012 (a 3-digit block); this further reduces to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/333⁠.</li> <li>0.999999... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/9⁠ = 1, since the repeating block is 9 (also a 1-digit block)</li></ul> If the repeating decimal is as above, except that there are k (extra) digits 0 between the decimal point and the repeating n-digit block, then one can simply add k digits 0 after the n digits 9 of the denominator (and, as before, the fraction may subsequently be simplified). For example, <ul><li>0.000444... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/9000⁠ since the repeating block is 4 and this block is preceded by 3 zeros,</li> <li>0.005656... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠56/9900⁠ since the repeating block is 56 and it is preceded by 2 zeros,</li> <li>0.00012012... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12/99900⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8325⁠ since the repeating block is 012 and it is preceded by 2 zeros.</li></ul> Any repeating decimal not of the form described above can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types (actually the first type suffices, but that could require the terminating decimal to be negative). For example, <ul><li>1.23444... = 1.23 + 0.00444... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠123/100⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/900⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1107/900⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/900⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1111/900⁠ <ul><li>or alternatively 1.23444... = 0.79 + 0.44444... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠79/100⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/9⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠711/900⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠400/900⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1111/900⁠</li></ul></li> <li>0.3789789... = 0.3 + 0.0789789... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/10⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠789/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2997/9990⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠789/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3786/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠631/1665⁠ <ul><li>or alternatively 0.3789789... = −0.6 + 0.9789789... = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/10⁠ + 978/999 = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5994/9990⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9780/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3786/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠631/1665⁠</li></ul></li></ul> An even faster method is to ignore the decimal point completely and go like this <ul><li>1.23444... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1234 − 123/900⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1111/900⁠ (denominator has one 9 and two 0s because one digit repeats and there are two non-repeating digits after the decimal point)</li> <li>0.3789789... = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3789 − 3/9990⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3786/9990⁠ (denominator has three 9s and one 0 because three digits repeat and there is one non-repeating digit after the decimal point)</li></ul> It follows that any repeating decimal with <a href="/wiki/Periodic_function" title="Periodic function">period</a> n, and k digits after the decimal point that do not belong to the repeating part, can be written as a (not necessarily reduced) fraction whose denominator is (10n − 1)10k. Conversely the period of the repeating decimal of a fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠c/d⁠ will be (at most) the smallest number n such that 10n − 1 is divisible by d. For example, the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/7⁠ has d = 7, and the smallest k that makes 10k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/7⁠ is therefore 6. <div class="mw-heading mw-heading4"><h4 id="In_compressed_form">In compressed form</h4>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=16" title="Edit section: In compressed form">edit</a>]</div> The following picture suggests kind of compression of the above shortcut. Thereby <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"> represents the digits of the integer part of the decimal number (to the left of the decimal point), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> makes up the string of digits of the preperiod and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f7b946ece1fc1bcb869894ba3796b7cd87ddfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.955ex; height:2.509ex;" alt="{\displaystyle \#\mathbf {A} }"> its length, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"> being the string of repeated digits (the period) with length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#\mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#\mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e06b55805eb98f69be9deefaccba44fe8cc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.763ex; height:2.509ex;" alt="{\displaystyle \#\mathbf {P} }"> which is nonzero. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CodeCogsEqn(4).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/CodeCogsEqn%284%29.gif/240px-CodeCogsEqn%284%29.gif" decoding="async" width="240" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/CodeCogsEqn%284%29.gif/360px-CodeCogsEqn%284%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/CodeCogsEqn%284%29.gif/480px-CodeCogsEqn%284%29.gif 2x" data-file-width="691" data-file-height="130" /></a><figcaption>Formation rule</figcaption></figure> In the generated fraction, the digit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d3d1e1f9dfe0254c628379e69a69711fe4eabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}"> will be repeated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#\mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#\mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e06b55805eb98f69be9deefaccba44fe8cc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.763ex; height:2.509ex;" alt="{\displaystyle \#\mathbf {P} }"> times, and the digit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> will be repeated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f7b946ece1fc1bcb869894ba3796b7cd87ddfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.955ex; height:2.509ex;" alt="{\displaystyle \#\mathbf {A} }"> times. Note that in the absence of an integer part in the decimal, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"> will be represented by zero, which being to the left of the other digits, will not affect the final result, and may be omitted in the calculation of the generating function. Examples: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lllll}3.254444\ldots &=3.25{\overline {4}}&={\begin{Bmatrix}\mathbf {I} =3&\mathbf {A} =25&\mathbf {P} =4\\&\#\mathbf {A} =2&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {3254-325}{900}}&={\dfrac {2929}{900}}\\\\0.512512\ldots &=0.{\overline {512}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =\emptyset &\mathbf {P} =512\\&\#\mathbf {A} =0&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {512-0}{999}}&={\dfrac {512}{999}}\\\\1.09191\ldots &=1.0{\overline {91}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =0&\mathbf {P} =91\\&\#\mathbf {A} =1&\#\mathbf {P} =2\end{Bmatrix}}&={\dfrac {1091-10}{990}}&={\dfrac {1081}{990}}\\\\1.333\ldots &=1.{\overline {3}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =\emptyset &\mathbf {P} =3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline {789}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =3&\mathbf {P} =789\\&\#\mathbf {A} =1&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {3789-3}{9990}}&={\dfrac {3786}{9990}}&={\dfrac {631}{1665}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left left left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3.254444</mn> <mo>…</mo> </mtd> <mtd> <mo>=</mo> <mn>3.25</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>4</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>25</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>2</mn> </mtd> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>3254</mn> <mo>−</mo> <mn>325</mn> </mrow> <mn>900</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>2929</mn> <mn>900</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>0.512512</mn> <mo>…</mo> </mtd> <mtd> <mo>=</mo> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>512</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>512</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>512</mn> <mo>−</mo> <mn>0</mn> </mrow> <mn>999</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>512</mn> <mn>999</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>1.09191</mn> <mo>…</mo> </mtd> <mtd> <mo>=</mo> <mn>1.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>91</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>91</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>1091</mn> <mo>−</mo> <mn>10</mn> </mrow> <mn>990</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1081</mn> <mn>990</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>1.333</mn> <mo>…</mo> </mtd> <mtd> <mo>=</mo> <mn>1.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>13</mn> <mo>−</mo> <mn>1</mn> </mrow> <mn>9</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>12</mn> <mn>9</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>0.3789789</mn> <mo>…</mo> </mtd> <mtd> <mo>=</mo> <mn>0.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>789</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>789</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>3789</mn> <mo>−</mo> <mn>3</mn> </mrow> <mn>9990</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>3786</mn> <mn>9990</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>631</mn> <mn>1665</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lllll}3.254444\ldots &=3.25{\overline {4}}&={\begin{Bmatrix}\mathbf {I} =3&\mathbf {A} =25&\mathbf {P} =4\\&\#\mathbf {A} =2&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {3254-325}{900}}&={\dfrac {2929}{900}}\\\\0.512512\ldots &=0.{\overline {512}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =\emptyset &\mathbf {P} =512\\&\#\mathbf {A} =0&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {512-0}{999}}&={\dfrac {512}{999}}\\\\1.09191\ldots &=1.0{\overline {91}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =0&\mathbf {P} =91\\&\#\mathbf {A} =1&\#\mathbf {P} =2\end{Bmatrix}}&={\dfrac {1091-10}{990}}&={\dfrac {1081}{990}}\\\\1.333\ldots &=1.{\overline {3}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =\emptyset &\mathbf {P} =3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline {789}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =3&\mathbf {P} =789\\&\#\mathbf {A} =1&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {3789-3}{9990}}&={\dfrac {3786}{9990}}&={\dfrac {631}{1665}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2073c8d83a97adc3816231d39d040cbc9a26e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.005ex; width:97.834ex; height:45.176ex;" alt="{\displaystyle {\begin{array}{lllll}3.254444\ldots &=3.25{\overline {4}}&={\begin{Bmatrix}\mathbf {I} =3&\mathbf {A} =25&\mathbf {P} =4\\&\#\mathbf {A} =2&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {3254-325}{900}}&={\dfrac {2929}{900}}\\\\0.512512\ldots &=0.{\overline {512}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =\emptyset &\mathbf {P} =512\\&\#\mathbf {A} =0&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {512-0}{999}}&={\dfrac {512}{999}}\\\\1.09191\ldots &=1.0{\overline {91}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =0&\mathbf {P} =91\\&\#\mathbf {A} =1&\#\mathbf {P} =2\end{Bmatrix}}&={\dfrac {1091-10}{990}}&={\dfrac {1081}{990}}\\\\1.333\ldots &=1.{\overline {3}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =\emptyset &\mathbf {P} =3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline {789}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =3&\mathbf {P} =789\\&\#\mathbf {A} =1&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {3789-3}{9990}}&={\dfrac {3786}{9990}}&={\dfrac {631}{1665}}\end{array}}}"> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> in the examples above denotes the absence of digits of part <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> in the decimal, and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#\mathbf {A} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#\mathbf {A} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6655dcdc7daf1eaeeb38ad8ae80ad76db6e2b3d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.216ex; height:2.509ex;" alt="{\displaystyle \#\mathbf {A} =0}"> and a corresponding absence in the generated fraction. <div class="mw-heading mw-heading2"><h2 id="Repeating_decimals_as_infinite_series">Repeating decimals as infinite series</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=17" title="Edit section: Repeating decimals as infinite series">edit</a>]</div> A repeating decimal can also be expressed as an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>. That is, a repeating decimal can be regarded as the sum of an infinite number of rational numbers. To take the simplest example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.{\overline {1}}={\frac {1}{10}}+{\frac {1}{100}}+{\frac {1}{1000}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>100</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1000</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.{\overline {1}}={\frac {1}{10}}+{\frac {1}{100}}+{\frac {1}{1000}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2238fe61d70636da4f6b5666194b2f95af719fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.62ex; height:6.843ex;" alt="{\displaystyle 0.{\overline {1}}={\frac {1}{10}}+{\frac {1}{100}}+{\frac {1}{1000}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}}"></dd></dl> The above series is a <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> with the first term as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/10⁠ and the common factor <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/10⁠. Because the absolute value of the common factor is less than 1, we can say that the geometric series <a href="/wiki/Convergent_series" title="Convergent series">converges</a> and find the exact value in the form of a fraction by using the following formula where a is the first term of the series and r is the common factor. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{1-r}}={\frac {\frac {1}{10}}{1-{\frac {1}{10}}}}={\frac {1}{10-1}}={\frac {1}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>10</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{1-r}}={\frac {\frac {1}{10}}{1-{\frac {1}{10}}}}={\frac {1}{10-1}}={\frac {1}{9}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d4555a6d732464008a18b6225cc16a732a5eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.665ex; height:8.176ex;" alt="{\displaystyle {\frac {a}{1-r}}={\frac {\frac {1}{10}}{1-{\frac {1}{10}}}}={\frac {1}{10-1}}={\frac {1}{9}}}"></dd></dl> Similarly, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}0.{\overline {142857}}&={\frac {142857}{10^{6}}}+{\frac {142857}{10^{12}}}+{\frac {142857}{10^{18}}}+\cdots =\sum _{n=1}^{\infty }{\frac {142857}{10^{6n}}}\\[6px]\implies &\quad {\frac {a}{1-r}}={\frac {\frac {142857}{10^{6}}}{1-{\frac {1}{10^{6}}}}}={\frac {142857}{10^{6}-1}}={\frac {142857}{999999}}={\frac {1}{7}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>142857</mn> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mn>142857</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>142857</mn> <mn>999999</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0.{\overline {142857}}&={\frac {142857}{10^{6}}}+{\frac {142857}{10^{12}}}+{\frac {142857}{10^{18}}}+\cdots =\sum _{n=1}^{\infty }{\frac {142857}{10^{6n}}}\\[6px]\implies &\quad {\frac {a}{1-r}}={\frac {\frac {142857}{10^{6}}}{1-{\frac {1}{10^{6}}}}}={\frac {142857}{10^{6}-1}}={\frac {142857}{999999}}={\frac {1}{7}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b641179f87b6e5f31361e4a13892afeb2a0e19d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:62.077ex; height:17.343ex;" alt="{\displaystyle {\begin{aligned}0.{\overline {142857}}&={\frac {142857}{10^{6}}}+{\frac {142857}{10^{12}}}+{\frac {142857}{10^{18}}}+\cdots =\sum _{n=1}^{\infty }{\frac {142857}{10^{6n}}}\\[6px]\implies &\quad {\frac {a}{1-r}}={\frac {\frac {142857}{10^{6}}}{1-{\frac {1}{10^{6}}}}}={\frac {142857}{10^{6}-1}}={\frac {142857}{999999}}={\frac {1}{7}}\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Multiplication_and_cyclic_permutation">Multiplication and cyclic permutation</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=18" title="Edit section: Multiplication and cyclic permutation">edit</a>]</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/Transposable_integer" title="Transposable integer">Transposable integer</a></div> The cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cyclically permuted</a> when multiplied by certain numbers. For example, 102564 × 4 = 410256. 102564 is the repetend of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/39⁠ and 410256 the repetend of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠16/39⁠. <div class="mw-heading mw-heading2"><h2 id="Other_properties_of_repetend_lengths">Other properties of repetend lengths</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=19" title="Edit section: Other properties of repetend lengths">edit</a>]</div> Various properties of repetend lengths (periods) are given by Mitchell<a href="#cite_note-13">[13]</a> and Dickson.<a href="#cite_note-14">[14]</a> <ul><li>The period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ for integer k is always ≤ k − 1.</li> <li>If p is prime, the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ divides evenly into p − 1.</li> <li>If k is composite, the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ is strictly less than k − 1.</li> <li>The period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠c/k⁠, for c <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to k, equals the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠.</li> <li>If k = 2a·5bn where n > 1 and n is not divisible by 2 or 5, then the length of the transient of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ is max(a, b), and the period equals r, where r is the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of 10 mod n, that is the smallest integer such that 10r ≡ 1 (mod n).</li> <li>If p, p′, p″,... are distinct primes, then the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p p′ p″ ⋯⁠ equals the lowest common multiple of the periods of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p′⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p″⁠,....</li> <li>If k and k′ have no common prime factors other than 2 or 5, then the period of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k k′⁠ equals the least common multiple of the periods of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k′⁠.</li> <li>For prime p, if</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{period}}\left({\frac {1}{p}}\right)={\text{period}}\left({\frac {1}{p^{2}}}\right)=\cdots ={\text{period}}\left({\frac {1}{p^{m}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{period}}\left({\frac {1}{p}}\right)={\text{period}}\left({\frac {1}{p^{2}}}\right)=\cdots ={\text{period}}\left({\frac {1}{p^{m}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bb5bb609446628fd74c25bbd05f15c75e5e971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.204ex; height:6.176ex;" alt="{\displaystyle {\text{period}}\left({\frac {1}{p}}\right)={\text{period}}\left({\frac {1}{p^{2}}}\right)=\cdots ={\text{period}}\left({\frac {1}{p^{m}}}\right)}"></dd></dl></dd> <dd>for some m, but <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{period}}\left({\frac {1}{p^{m}}}\right)\neq {\text{period}}\left({\frac {1}{p^{m+1}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{period}}\left({\frac {1}{p^{m}}}\right)\neq {\text{period}}\left({\frac {1}{p^{m+1}}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4d1531be5792910cb0a865b8fedccf509c14d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.887ex; height:6.176ex;" alt="{\displaystyle {\text{period}}\left({\frac {1}{p^{m}}}\right)\neq {\text{period}}\left({\frac {1}{p^{m+1}}}\right),}"></dd></dl></dd> <dd>then for c ≥ 0 we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{period}}\left({\frac {1}{p^{m+c}}}\right)=p^{c}\cdot {\text{period}}\left({\frac {1}{p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>c</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>period</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{period}}\left({\frac {1}{p^{m+c}}}\right)=p^{c}\cdot {\text{period}}\left({\frac {1}{p}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099b983765f03e24219f12dca9170a694b7b34e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.895ex; height:6.176ex;" alt="{\displaystyle {\text{period}}\left({\frac {1}{p^{m+c}}}\right)=p^{c}\cdot {\text{period}}\left({\frac {1}{p}}\right).}"></dd></dl></dd></dl> <ul><li>If p is a proper prime ending in a 1, that is, if the repetend of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ is a cyclic number of length p − 1 and p = 10h + 1 for some h, then each digit 0, 1, ..., 9 appears in the repetend exactly h = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p − 1/10⁠ times.</li></ul> For some other properties of repetends, see also.<a href="#cite_note-15">[15]</a> <div class="mw-heading mw-heading2"><h2 id="Extension_to_other_bases">Extension to other bases</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=20" title="Edit section: Extension to other bases">edit</a>]</div> Various features of repeating decimals extend to the representation of numbers in all other integer bases, not just base 10: <ul><li>Every real number can be represented as an integer part followed by a <a href="/wiki/Radix" title="Radix">radix</a> point (the generalization of a <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> to non-decimal systems) followed by a finite or infinite number of <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a>.</li> <li>If the base is an integer, a terminating sequence obviously represents a rational number.</li> <li>A rational number has a terminating sequence if all the prime factors of the denominator of the fully reduced fractional form are also factors of the base. These numbers make up a <a href="/wiki/Dense_set" title="Dense set">dense set</a> in Q and R.</li> <li>If the <a href="/wiki/Positional_notation" title="Positional notation">positional numeral system</a> is a standard one, that is it has base</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \mathbb {Z} \smallsetminus \{-1,0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in \mathbb {Z} \smallsetminus \{-1,0,1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1498adef20545607bc265f16ab25d2b18a65e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.917ex; height:2.843ex;" alt="{\displaystyle b\in \mathbb {Z} \smallsetminus \{-1,0,1\}}"></dd></dl></dd> <dd>combined with a consecutive set of digits <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D:=\{d_{1},d_{1}+1,\dots ,d_{r}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D:=\{d_{1},d_{1}+1,\dots ,d_{r}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440159dc9938a423e96e62f245799e2d0bb89d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.918ex; height:2.843ex;" alt="{\displaystyle D:=\{d_{1},d_{1}+1,\dots ,d_{r}\}}"></dd></dl></dd> <dd>with r := |b|, dr := d1 + r − 1 and 0 ∈ D, then a terminating sequence is obviously equivalent to the same sequence with non-terminating repeating part consisting of the digit 0. If the base is positive, then there exists an <a href="/wiki/Order_isomorphism" title="Order isomorphism">order homomorphism</a> from the <a href="/wiki/String_(computer_science)#Lexicographical_ordering" title="String (computer science)">lexicographical order</a> of the <a href="/wiki/Sequence#Finite_and_infinite" title="Sequence">right-sided infinite strings</a> over the <a href="/wiki/Alphabet" title="Alphabet">alphabet</a> D into some closed interval of the reals, which maps the strings 0.A1A2...Andb and 0.A1A2...(An+1)d1 with Ai ∈ D and An ≠ db to the same real number – and there are no other duplicate images. In the decimal system, for example, there is 0.9 = 1.0 = 1; in the <a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a> system there is 0.1 = 1.T = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠.</dd></dl> <ul><li>A rational number has an indefinitely repeating sequence of finite length l, if the reduced fraction's denominator contains a prime factor that is not a factor of the base. If q is the maximal factor of the reduced denominator which is coprime to the base, l is the smallest exponent such that q divides bℓ − 1. It is the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> ordq(b) of the residue class b mod q which is a divisor of the <a href="/wiki/Carmichael_function" title="Carmichael function">Carmichael function</a> λ(q) which in turn is smaller than q. The repeating sequence is preceded by a transient of finite length if the reduced fraction also shares a prime factor with the base. A repeating sequence</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0.{\overline {A_{1}A_{2}\ldots A_{\ell }}}\right)_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(0.{\overline {A_{1}A_{2}\ldots A_{\ell }}}\right)_{b}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e49c18417a4fd4884c5df63e704a6dd707e4689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.391ex; height:4.843ex;" alt="{\displaystyle \left(0.{\overline {A_{1}A_{2}\ldots A_{\ell }}}\right)_{b}}"></dd></dl></dd> <dd>represents the fraction <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(A_{1}A_{2}\ldots A_{\ell })_{b}}{b^{\ell }-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(A_{1}A_{2}\ldots A_{\ell })_{b}}{b^{\ell }-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c36bd49309e777c2a330928c3c14e60f5b335f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.983ex; height:6.176ex;" alt="{\displaystyle {\frac {(A_{1}A_{2}\ldots A_{\ell })_{b}}{b^{\ell }-1}}.}"></dd></dl></dd></dl> <ul><li>An irrational number has a representation of infinite length that is not, from any point, an indefinitely repeating sequence of finite length.</li></ul> For example, in <a href="/wiki/Duodecimal" title="Duodecimal">duodecimal</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ = 0.6, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠ = 0.4, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ = 0.3 and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ = 0.2 all terminate; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/5⁠ = 0.2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ = 0.186A35 has period 6 in duodecimal, just as it does in decimal. If b is an integer base and k is an integer, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{k}}={\frac {1}{b}}+{\frac {(b-k)^{1}}{b^{2}}}+{\frac {(b-k)^{2}}{b^{3}}}+{\frac {(b-k)^{3}}{b^{4}}}+\cdots +{\frac {(b-k)^{N-1}}{b^{N}}}+\cdots ={\frac {1}{b}}{\frac {1}{1-{\frac {b-k}{b}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>k</mi> </mrow> <mi>b</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{k}}={\frac {1}{b}}+{\frac {(b-k)^{1}}{b^{2}}}+{\frac {(b-k)^{2}}{b^{3}}}+{\frac {(b-k)^{3}}{b^{4}}}+\cdots +{\frac {(b-k)^{N-1}}{b^{N}}}+\cdots ={\frac {1}{b}}{\frac {1}{1-{\frac {b-k}{b}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7809fcadf4e6d09289a3d0dd18fada41634f5428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:81.626ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{k}}={\frac {1}{b}}+{\frac {(b-k)^{1}}{b^{2}}}+{\frac {(b-k)^{2}}{b^{3}}}+{\frac {(b-k)^{3}}{b^{4}}}+\cdots +{\frac {(b-k)^{N-1}}{b^{N}}}+\cdots ={\frac {1}{b}}{\frac {1}{1-{\frac {b-k}{b}}}}.}"></dd></dl> For example 1/7 in duodecimal: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{7}}=\left({\frac {1}{10^{\phantom {1}}}}+{\frac {5}{10^{2}}}+{\frac {21}{10^{3}}}+{\frac {A5}{10^{4}}}+{\frac {441}{10^{5}}}+{\frac {1985}{10^{6}}}+\cdots \right)_{\text{base 12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>1</mn> </mphantom> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mn>5</mn> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>441</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1985</mn> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>base 12</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{7}}=\left({\frac {1}{10^{\phantom {1}}}}+{\frac {5}{10^{2}}}+{\frac {21}{10^{3}}}+{\frac {A5}{10^{4}}}+{\frac {441}{10^{5}}}+{\frac {1985}{10^{6}}}+\cdots \right)_{\text{base 12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c615701a873b22f0a1008e536232b488cc8364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.525ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{7}}=\left({\frac {1}{10^{\phantom {1}}}}+{\frac {5}{10^{2}}}+{\frac {21}{10^{3}}}+{\frac {A5}{10^{4}}}+{\frac {441}{10^{5}}}+{\frac {1985}{10^{6}}}+\cdots \right)_{\text{base 12}}}"> which is 0.186A35base12. 10base12 is 12base10, 102base12 is 144base10, 21base12 is 25base10, A5base12 is 125base10. <div class="mw-heading mw-heading3"><h3 id="Algorithm_for_positive_bases">Algorithm for positive bases</h3>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=21" title="Edit section: Algorithm for positive bases">edit</a>]</div> For a rational 0 < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p/q⁠ < 1 (and base b ∈ N>1) there is the following algorithm producing the repetend together with its length: <div class="mw-highlight mw-highlight-lang-mupad mw-content-ltr" dir="ltr"><pre>function b_adic(b,p,q) // b ≥ 2; 0 digits = "0123..."; // up to the digit with value b–1 begin s = ""; // the string of digits pos = 0; // all places are right to the radix point while not defined(occurs[p]) do occurs[p] = pos; // the position of the place with remainder p bp = b*p; z = floor(bp/q); // index z of digit within: 0 ≤ z ≤ b-1 p = b*p − z*q; // 0 ≤ p < q if p = 0 then L = 0; if not z = 0 then s = s . substring(digits, z, 1) end if return (s); end if s = s . substring(digits, z, 1); // append the character of the digit pos += 1; end while L = pos - occurs[p]; // the length of the repetend (being < q) // mark the digits of the repetend by a vinculum: for i from occurs[p] to pos-1 do substring(s, i, 1) = overline(substring(s, i, 1)); end for return (s); end function </pre></div> The first highlighted line calculates the digit z. The subsequent line calculates the new remainder p′ of the division <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> the denominator q. As a consequence of the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor function</a> <code>floor</code> we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {bp}{q}}-1\;\;<\;\;z=\left\lfloor {\frac {bp}{q}}\right\rfloor \;\;\leq \;\;{\frac {bp}{q}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>p</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo><</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>p</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>≤</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>p</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {bp}{q}}-1\;\;<\;\;z=\left\lfloor {\frac {bp}{q}}\right\rfloor \;\;\leq \;\;{\frac {bp}{q}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee4587b4380cef1fb47c962cbf4bd144c0e274b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.914ex; height:6.176ex;" alt="{\displaystyle {\frac {bp}{q}}-1\;\;<\;\;z=\left\lfloor {\frac {bp}{q}}\right\rfloor \;\;\leq \;\;{\frac {bp}{q}},}"></dd></dl> thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bp-q<zq\quad \implies \quad p':=bp-zq<q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>p</mi> <mo>−</mo> <mi>q</mi> <mo><</mo> <mi>z</mi> <mi>q</mi> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>:=</mo> <mi>b</mi> <mi>p</mi> <mo>−</mo> <mi>z</mi> <mi>q</mi> <mo><</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bp-q<zq\quad \implies \quad p':=bp-zq<q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0518b0f5ba3c328642bb709181043914cf06d9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.297ex; height:2.843ex;" alt="{\displaystyle bp-q<zq\quad \implies \quad p':=bp-zq<q}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zq\leq bp\quad \implies \quad 0\leq bp-zq=:p'\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>q</mi> <mo>≤</mo> <mi>b</mi> <mi>p</mi> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>b</mi> <mi>p</mi> <mo>−</mo> <mi>z</mi> <mi>q</mi> <mo>=:</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zq\leq bp\quad \implies \quad 0\leq bp-zq=:p'\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa2c2e589c10809a602f66f0dd634f1b692a91b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.514ex; height:2.843ex;" alt="{\displaystyle zq\leq bp\quad \implies \quad 0\leq bp-zq=:p'\,.}"></dd></dl> Because all these remainders p are non-negative integers less than q, there can be only a finite number of them with the consequence that they must recur in the <code>while</code> loop. Such a recurrence is detected by the <a href="/wiki/Associative_array" title="Associative array">associative array</a> <code>occurs</code>. The new digit z is formed in the yellow line, where p is the only non-constant. The length L of the repetend equals the number of the remainders (see also section <a href="#Every_rational_number_is_either_a_terminating_or_repeating_decimal">Every rational number is either a terminating or repeating decimal</a>). <div class="mw-heading mw-heading2"><h2 id="Applications_to_cryptography">Applications to cryptography</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=22" title="Edit section: Applications to cryptography">edit</a>]</div> Repeating decimals (also called decimal sequences) have found cryptographic and error-correction coding applications.<a href="#cite_note-16">[16]</a> In these applications repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/p⁠ (when 2 is a primitive root of p) is given by:<a href="#cite_note-17">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45043b161b23bd27f94c30775e6d9633bd58ef8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.596ex; height:3.176ex;" alt="{\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}}"></dd></dl> These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p − 1/2⁠. The randomness of these sequences has been examined by <a href="/wiki/Diehard_tests" title="Diehard tests">diehard tests</a>.<a href="#cite_note-18">[18]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=23" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Decimal_representation" title="Decimal representation">Decimal representation</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend prime</a></li> <li><a href="/wiki/Midy%27s_theorem" title="Midy's theorem">Midy's theorem</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic number</a></li> <li><a href="/wiki/Trailing_zero" title="Trailing zero">Trailing zero</a></li> <li><a href="/wiki/Unique_prime" class="mw-redirect" title="Unique prime">Unique prime</a></li> <li><a href="/wiki/0.999..." title="0.999...">0.999...</a>, a repeating decimal equal to one</li> <li><a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">Pigeonhole principle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=24" title="Edit section: Notes">edit</a>]</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-1"><a href="#cite_ref-1">^</a> Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996: p. 67. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBeswick2004" class="citation cs2">Beswick, Kim (2004), "Why Does 0.999... = 1?: A Perennial Question and Number Sense", Australian Mathematics Teacher, 60 (4): 7–9</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Australian+Mathematics+Teacher&rft.atitle=Why+Does+0.999...+%3D+1%3F%3A+A+Perennial+Question+and+Number+Sense&rft.volume=60&rft.issue=4&rft.pages=7-9&rft.date=2004&rft.aulast=Beswick&rft.aufirst=Kim&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/895611/lamberts-original-proof-that-pi-is-irrational">"Lambert's Original Proof that $\pi$ is irrational"</a>. Mathematics Stack Exchange. Retrieved 2023-12-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematics+Stack+Exchange&rft.atitle=Lambert%27s+Original+Proof+that+%24%5Cpi%24+is+irrational.&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F895611%2Flamberts-original-proof-that-pi-is-irrational&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011). Aide-mémoire. Mathématiques 9-10-11. LEP. pp. 20–21. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> For a base b and a divisor n, in terms of group theory <a href="/wiki/Carmichael_function#Order_of_elements_modulo_n" title="Carmichael function">this length</a> divides <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ord} _{n}(b):=\min\{L\in \mathbb {N} \,\mid \,b^{L}\equiv 1{\bmod {n}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ord</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>L</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thinmathspace" /> <mo>∣</mo> <mspace width="thinmathspace" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} _{n}(b):=\min\{L\in \mathbb {N} \,\mid \,b^{L}\equiv 1{\bmod {n}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c89f02c770bea80606bb6ad11a7fb832c81083f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.836ex; height:3.176ex;" alt="{\displaystyle \operatorname {ord} _{n}(b):=\min\{L\in \mathbb {N} \,\mid \,b^{L}\equiv 1{\bmod {n}}\}}"></dd></dl> (with <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> ≡ 1 mod n) which divides the Carmichael function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,n)=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ord</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∣</mo> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,n)=1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a96b36d6386d94b9865d41df7076a48dd7b003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.042ex; height:2.843ex;" alt="{\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,n)=1\}}"></dd></dl> which again divides <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a> φ(n). </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVuorinen" class="citation web cs1">Vuorinen, Aapeli. <a rel="nofollow" class="external text" href="https://www.aapelivuorinen.com/blog/2017/03/06/rational-numbers-repeating-decimal-expansions/">"Rational numbers have repeating decimal expansions"</a>. Aapeli Vuorinen. Retrieved 2023-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Aapeli+Vuorinen&rft.atitle=Rational+numbers+have+repeating+decimal+expansions&rft.aulast=Vuorinen&rft.aufirst=Aapeli&rft_id=https%3A%2F%2Fwww.aapelivuorinen.com%2Fblog%2F2017%2F03%2F06%2Frational-numbers-repeating-decimal-expansions%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20231223234518/https://www.sjsu.edu/faculty/watkins/repeatingdecimals.htm">"The Sets of Repeating Decimals"</a>. www.sjsu.edu. Archived from <a rel="nofollow" class="external text" href="https://www.sjsu.edu/faculty/watkins/repeatingdecimals.htm">the original</a> on 23 December 2023. Retrieved 2023-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.sjsu.edu&rft.atitle=The+Sets+of+Repeating+Decimals&rft_id=https%3A%2F%2Fwww.sjsu.edu%2Ffaculty%2Fwatkins%2Frepeatingdecimals.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoRi2016" class="citation web cs1">RoRi (2016-03-01). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231223234333/https://www.stumblingrobot.com/2016/02/29/prove-that-every-repeating-decimal-represents-a-rational-number/">"Prove that every repeating decimal represents a rational number"</a>. Stumbling Robot. Archived from <a rel="nofollow" class="external text" href="https://www.stumblingrobot.com/2016/02/29/prove-that-every-repeating-decimal-represents-a-rational-number/">the original</a> on 23 December 2023. Retrieved 2023-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stumbling+Robot&rft.atitle=Prove+that+every+repeating+decimal+represents+a+rational+number&rft.date=2016-03-01&rft.au=RoRi&rft_id=https%3A%2F%2Fwww.stumblingrobot.com%2F2016%2F02%2F29%2Fprove-that-every-repeating-decimal-represents-a-rational-number%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGray2000" class="citation journal cs1">Gray, Alexander J. (March 2000). "Digital roots and reciprocals of primes". <a href="/wiki/Mathematical_Gazette" class="mw-redirect" title="Mathematical Gazette">Mathematical Gazette</a>. 84 (499): 86. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3621484">10.2307/3621484</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3621484">3621484</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125834304">125834304</a>. <q>For primes greater than 5, all the digital roots appear to have the same value, 9. We can confirm this if...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Gazette&rft.atitle=Digital+roots+and+reciprocals+of+primes&rft.volume=84&rft.issue=499&rft.pages=86&rft.date=2000-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125834304%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3621484%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3621484&rft.aulast=Gray&rft.aufirst=Alexander+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> William E. Heal. Some Properties of Repetends. Annals of Mathematics, Vol. 3, No. 4 (Aug., 1887), pp. 97–103 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Albert H. Beiler, Recreations in the Theory of Numbers, p. 79 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length", <a href="/wiki/Cryptologia" title="Cryptologia">Cryptologia</a> 17, January 1993, pp. 55–62. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="/wiki/L._E._Dickson" class="mw-redirect" title="L. E. Dickson">Dickson, Leonard E.</a>, <a href="/wiki/History_of_the_Theory_of_Numbers" title="History of the Theory of Numbers">History of the Theory of Numbers</a>, Vol. I, Chelsea Publ. Co., 1952 (orig. 1918), pp. 164–173. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Armstrong, N. J., and Armstrong, R. J., "Some properties of repetends", Mathematical Gazette 87, November 2003, pp. 437–443. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Kak, Subhash, Chatterjee, A. "On decimal sequences". IEEE Transactions on Information Theory, vol. IT-27, pp. 647–652, September 1981. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Kak, Subhash, "Encryption and error-correction using d-sequences". IEEE Transactios on Computers, vol. C-34, pp. 803–809, 1985. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Bellamy, J. "Randomness of D sequences via diehard testing". 2013. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1312.3618">1312.3618</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Repeating_decimal&action=edit&section=25" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RepeatingDecimal.html">"Repeating Decimal"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Repeating+Decimal&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRepeatingDecimal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARepeating+decimal" class="Z3988"></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Repeating_decimal&oldid=1258671238">https://en.wikipedia.org/w/index.php?title=Repeating_decimal&oldid=1258671238</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Elementary_arithmetic" title="Category:Elementary arithmetic">Elementary arithmetic</a></li><li><a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">Numeral systems</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_July_2023" title="Category:Articles needing additional references from July 2023">Articles needing additional references from July 2023</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_cleanup_from_July_2024" title="Category:Articles needing cleanup from July 2024">Articles needing cleanup from July 2024</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Cleanup_tagged_articles_with_a_reason_field_from_July_2024" title="Category:Cleanup tagged articles with a reason field from July 2024">Cleanup tagged articles with a reason field from July 2024</a></li><li><a href="/wiki/Category:Wikipedia_pages_needing_cleanup_from_July_2024" title="Category:Wikipedia pages needing cleanup from July 2024">Wikipedia pages needing cleanup from July 2024</a></li><li><a href="/wiki/Category:Articles_with_multiple_maintenance_issues" title="Category:Articles with multiple maintenance issues">Articles with multiple maintenance issues</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2022" title="Category:Articles needing additional references from May 2022">Articles needing additional references from May 2022</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_October_2023" title="Category:Articles with unsourced statements from October 2023">Articles with unsourced statements from October 2023</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 20 November 2024, at 23:11 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Repeating_decimal&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-m6fsj","wgBackendResponseTime":155,"wgPageParseReport":{"limitreport":{"cputime":"0.832","walltime":"1.056","ppvisitednodes":{"value":13196,"limit":1000000},"postexpandincludesize":{"value":191442,"limit":2097152},"templateargumentsize":{"value":22411,"limit":2097152},"expansiondepth":{"value":20,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":208687,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 719.729 1 -total"," 24.88% 179.082 243 Template:Sfrac"," 16.26% 117.047 1 Template:Reflist"," 13.32% 95.900 1 Template:Multiple_issues"," 10.95% 78.801 1 Template:Citation"," 10.26% 73.833 1 Template:Short_description"," 8.54% 61.476 3 Template:Ambox"," 7.49% 53.890 2 Template:Pagetype"," 7.20% 51.815 1 Template:More_citations_needed"," 4.55% 32.767 4 Template:Cn"]},"scribunto":{"limitreport-timeusage":{"value":"0.308","limit":"10.000"},"limitreport-memusage":{"value":6513421,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-2p7sq","timestamp":"20241202004516","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Repeating decimal","url":"https:\/\/en.wikipedia.org\/wiki\/Repeating_decimal","sameAs":"http:\/\/www.wikidata.org\/entity\/Q389208","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q389208","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-02-06T19:24:23Z","dateModified":"2024-11-20T23:11:52Z","headline":"decimal representation of a number whose digits are periodic"}</script> </body> </html>

CINXE.COM

Repeating decimal - Wikipedia