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numbers</span> </div> </a> <ul id="toc-Approximation_using_decimal_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Infinite_decimal_expansion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infinite_decimal_expansion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Infinite decimal expansion</span> </div> </a> <button aria-controls="toc-Infinite_decimal_expansion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Infinite decimal expansion subsection</span> </button> <ul id="toc-Infinite_decimal_expansion-sublist" class="vector-toc-list"> <li id="toc-Rational_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rational numbers</span> </div> </a> <ul id="toc-Rational_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decimal_computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Decimal_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Decimal computation</span> </div> </a> <ul id="toc-Decimal_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-History_of_decimal_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_of_decimal_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>History of decimal fractions</span> </div> </a> <ul id="toc-History_of_decimal_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natural_languages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Natural languages</span> </div> </a> <ul id="toc-Natural_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_bases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_bases"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Other bases</span> </div> </a> <ul id="toc-Other_bases-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </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"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Decimal</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 86 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-86" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">86 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Dezimalsystem" title="Dezimalsystem – Alemannic" lang="gsw" hreflang="gsw" data-title="Dezimalsystem" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%B9%D8%AF_%D8%B9%D8%B4%D8%B1%D9%8A" title="نظام عد عشري – Arabic" lang="ar" hreflang="ar" data-title="نظام عد عشري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Sistema_de_numberaci%C3%B3n_decimal" title="Sistema de numberación decimal – Asturian" lang="ast" hreflang="ast" data-title="Sistema de numberación decimal" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Onluq_sistemi" title="Onluq sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Onluq sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A6%B6%E0%A6%AE%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BF" title="দশমিক পদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="দশমিক পদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%CC%8Dp-ch%C3%ACn-hoat" title="Si̍p-chìn-hoat – Minnan" lang="nan" hreflang="nan" data-title="Si̍p-chìn-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B0%D1%80%D0%BB%D1%8B_%D0%B8%D2%AB%D3%99%D0%BF%D0%BB%D3%99%D2%AF_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Унарлы иҫәпләү системаһы – Bashkir" lang="ba" hreflang="ba" data-title="Унарлы иҫәпләү системаһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D1%81%D1%8F%D1%82%D0%BA%D0%BE%D0%B2%D0%B0%D1%8F_%D1%81%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B7%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D0%BD%D1%8F" title="Дзесятковая сістэма злічэння – Belarusian" lang="be" hreflang="be" data-title="Дзесятковая сістэма злічэння" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D1%81%D1%8F%D1%82%D0%BA%D0%BE%D0%B2%D0%B0%D1%8F_%D1%81%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B7%D1%8C%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D1%8C%D0%BD%D1%8F" title="Дзесятковая сыстэма зьлічэньня – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дзесятковая сыстэма зьлічэньня" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Desimal" title="Desimal – Central Bikol" lang="bcl" hreflang="bcl" data-title="Desimal" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B1%D1%80%D0%BE%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Десетична бройна система – Bulgarian" lang="bg" hreflang="bg" data-title="Десетична бройна система" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Dekadni_sistem_brojeva" title="Dekadni sistem brojeva – Bosnian" lang="bs" hreflang="bs" data-title="Dekadni sistem brojeva" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Dispakadur_dekredel" title="Dispakadur dekredel – Breton" lang="br" hreflang="br" data-title="Dispakadur dekredel" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_decimal" title="Nombre decimal – Catalan" lang="ca" hreflang="ca" data-title="Nombre decimal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D1%83%D0%BD%D1%88%D0%B0%D1%80%D0%BB%C4%83_%D1%88%D1%83%D1%82_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Вуншарлă шут тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Вуншарлă шут тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Des%C3%ADtkov%C3%A1_soustava" title="Desítková soustava – Czech" lang="cs" hreflang="cs" data-title="Desítková soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Muravanegumi" title="Muravanegumi – Shona" lang="sn" hreflang="sn" data-title="Muravanegumi" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Degol" title="Degol – Welsh" lang="cy" hreflang="cy" data-title="Degol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dezimalsystem" title="Dezimalsystem – German" lang="de" hreflang="de" data-title="Dezimalsystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/K%C3%BCmnends%C3%BCsteem" title="Kümnendsüsteem – Estonian" lang="et" hreflang="et" data-title="Kümnendsüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B5%CE%BA%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1" title="Δεκαδικό σύστημα – Greek" lang="el" hreflang="el" data-title="Δεκαδικό σύστημα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_de_numeraci%C3%B3n_decimal" title="Sistema de numeración decimal – Spanish" lang="es" hreflang="es" data-title="Sistema de numeración decimal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dekuma_nombrosistemo" title="Dekuma nombrosistemo – Esperanto" lang="eo" hreflang="eo" data-title="Dekuma nombrosistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki-sistema_hamartar" title="Zenbaki-sistema hamartar – Basque" lang="eu" hreflang="eu" data-title="Zenbaki-sistema hamartar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%87%E2%80%8C%D8%AF%D9%87%DB%8C" title="ده‌دهی – Persian" lang="fa" hreflang="fa" data-title="ده‌دهی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_d%C3%A9cimal" title="Système décimal – French" lang="fr" hreflang="fr" data-title="Système décimal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%B3ras_deach%C3%BAil" title="Córas deachúil – Irish" lang="ga" hreflang="ga" data-title="Córas deachúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_numeraci%C3%B3n_decimal" title="Sistema de numeración decimal – Galician" lang="gl" hreflang="gl" data-title="Sistema de numeración decimal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%AD%EC%A7%84%EB%B2%95" title="십진법 – Korean" lang="ko" hreflang="ko" data-title="십진법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A4%B6%E0%A4%AE%E0%A4%B2%E0%A4%B5_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="दशमलव पद्धति – Hindi" lang="hi" hreflang="hi" data-title="दशमलव पद्धति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dekadski_brojevni_sustav" title="Dekadski brojevni sustav – Croatian" lang="hr" hreflang="hr" data-title="Dekadski brojevni sustav" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Decimala_nombrosistemo" title="Decimala nombrosistemo – Ido" lang="io" hreflang="io" data-title="Decimala nombrosistemo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_bilangan_desimal" title="Sistem bilangan desimal – Indonesian" lang="id" hreflang="id" data-title="Sistem bilangan desimal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-decimal" title="I-decimal – Xhosa" lang="xh" hreflang="xh" data-title="I-decimal" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tugakerfi" title="Tugakerfi – Icelandic" lang="is" hreflang="is" data-title="Tugakerfi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_numerico_decimale" title="Sistema numerico decimale – Italian" lang="it" hreflang="it" data-title="Sistema numerico decimale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A9%D7%99%D7%98%D7%94_%D7%94%D7%A2%D7%A9%D7%A8%D7%95%D7%A0%D7%99%D7%AA" title="השיטה העשרונית – Hebrew" lang="he" hreflang="he" data-title="השיטה העשרונית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Sistem_wilangan_d%C3%A8simal" title="Sistem wilangan dèsimal – Javanese" lang="jv" hreflang="jv" data-title="Sistem wilangan dèsimal" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B2%B6%E0%B2%AE%E0%B2%BE%E0%B2%A8_%E0%B2%AA%E0%B2%A6%E0%B3%8D%E0%B2%A7%E0%B2%A4%E0%B2%BF" title="ದಶಮಾನ ಪದ್ಧತಿ – Kannada" lang="kn" hreflang="kn" data-title="ದಶಮಾನ ಪದ್ಧತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D0%BD%D0%B4%D1%8B%D2%9B_%D1%81%D0%B0%D0%BD%D0%B0%D1%83_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Ондық санау жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="Ондық санау жүйесі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_desimal" title="Sistèm desimal – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm desimal" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9E%D0%BD%D0%B4%D1%83%D0%BA_%D1%8D%D1%81%D0%B5%D0%BF%D1%82%D3%A9%D3%A9_%D1%82%D1%83%D1%82%D1%83%D0%BC%D1%83" title="Ондук эсептөө тутуму – Kyrgyz" lang="ky" hreflang="ky" data-title="Ондук эсептөө тутуму" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_numericum_decimale" title="Systema numericum decimale – Latin" lang="la" hreflang="la" data-title="Systema numericum decimale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Decim%C4%81l%C4%81_skait%C4%AB%C5%A1anas_sist%C4%93ma" title="Decimālā skaitīšanas sistēma – Latvian" lang="lv" hreflang="lv" data-title="Decimālā skaitīšanas sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/De%C5%A1imtain%C4%97_skai%C4%8Diavimo_sistema" title="Dešimtainė skaičiavimo sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Dešimtainė skaičiavimo sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sistem_desimal" title="Sistem desimal – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Sistem desimal" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%ADzes_sz%C3%A1mrendszer" title="Tízes számrendszer – Hungarian" lang="hu" hreflang="hu" data-title="Tízes számrendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Десетичен броен систем – Macedonian" lang="mk" hreflang="mk" data-title="Десетичен броен систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B4%B6%E0%B4%BE%E0%B4%82%E0%B4%B6_%E0%B4%B8%E0%B4%AE%E0%B5%8D%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A6%E0%B4%BE%E0%B4%AF%E0%B4%82" title="ദശാംശ സമ്പ്രദായം – Malayalam" lang="ml" hreflang="ml" data-title="ദശാംശ സമ്പ്രദായം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Perpuluhan" title="Perpuluhan – Malay" lang="ms" hreflang="ms" data-title="Perpuluhan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sistema_de_numera%C3%A7on_decimal" title="Sistema de numeraçon decimal – Mirandese" lang="mwl" hreflang="mwl" data-title="Sistema de numeraçon decimal" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D0%B2%D1%82%D1%8B%D0%BD_%D1%82%D0%BE%D0%BE%D0%BB%D0%BB%D1%8B%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Аравтын тооллын систем – Mongolian" lang="mn" hreflang="mn" data-title="Аравтын тооллын систем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Decimaal_talstelsel" title="Decimaal talstelsel – Dutch" lang="nl" hreflang="nl" data-title="Decimaal talstelsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%81%E9%80%B2%E6%B3%95" title="十進法 – Japanese" lang="ja" hreflang="ja" data-title="十進法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Titallsystemet" title="Titallsystemet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Titallsystemet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Titalssystemet" title="Titalssystemet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Titalssystemet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_decimal" title="Sistèma decimal – Occitan" lang="oc" hreflang="oc" data-title="Sistèma decimal" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9B%D1%83%D0%B0%D0%BD_%D1%87%D0%BE%D1%82%D1%80%D0%B0%D0%B4%D0%B0%D0%BC_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B5" title="Луан чотрадам системе – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Луан чотрадам системе" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%87%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%B0%E0%A9%80%E0%A8%86" title="ਇਸ਼ਾਰੀਆ – Punjabi" lang="pa" hreflang="pa" data-title="ਇਸ਼ਾਰੀਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dziesi%C4%99tny_system_liczbowy" title="Dziesiętny system liczbowy – Polish" lang="pl" hreflang="pl" data-title="Dziesiętny system liczbowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_numera%C3%A7%C3%A3o_decimal" title="Sistema de numeração decimal – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de numeração decimal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_zecimal" title="Sistem zecimal – Romanian" lang="ro" hreflang="ro" data-title="Sistem zecimal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chunkantin_huchha_llika" title="Chunkantin huchha llika – Quechua" lang="qu" hreflang="qu" data-title="Chunkantin huchha llika" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D1%8F%D1%82%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Десятичная система счисления – Russian" lang="ru" hreflang="ru" data-title="Десятичная система счисления" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Letlase_la_lesome" title="Letlase la lesome – Northern Sotho" lang="nso" hreflang="nso" data-title="Letlase la lesome" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Northern Sotho" class="interlanguage-link-target"><span>Sesotho sa Leboa</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Decimal" title="Decimal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Decimal" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Desiatkov%C3%A1_%C4%8D%C3%ADseln%C3%A1_s%C3%BAstava" title="Desiatková číselná sústava – Slovak" lang="sk" hreflang="sk" data-title="Desiatková číselná sústava" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Deseti%C5%A1ki_%C5%A1tevilski_sistem" title="Desetiški številski sistem – Slovenian" lang="sl" hreflang="sl" data-title="Desetiški številski sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%DB%95%D8%AF%DB%95%DB%8C%DB%8C" title="دەدەیی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دەدەیی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D0%B4%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Декадни систем – Serbian" lang="sr" hreflang="sr" data-title="Декадни систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dekadni_sistem" title="Dekadni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Dekadni sistem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kymmenj%C3%A4rjestelm%C3%A4" title="Kymmenjärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Kymmenjärjestelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Decimala_talsystemet" title="Decimala talsystemet – Swedish" lang="sv" hreflang="sv" data-title="Decimala talsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AE%AE%E0%AF%8D" title="பதின்மம் – Tamil" lang="ta" hreflang="ta" data-title="பதின்மம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B0%D1%80%D0%BB%D1%8B_%D0%B8%D1%81%D3%99%D0%BF%D0%BB%D3%99%D2%AF_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Унарлы исәпләү системасы – Tatar" lang="tt" hreflang="tt" data-title="Унарлы исәпләү системасы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%AA%E0%B8%B4%E0%B8%9A" title="เลขฐานสิบ – Thai" lang="th" hreflang="th" data-title="เลขฐานสิบ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Onlu_say%C4%B1_sistemi" title="Onlu sayı sistemi – Turkish" lang="tr" hreflang="tr" data-title="Onlu sayı sistemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D1%8F%D1%82%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Десяткова система числення – Ukrainian" lang="uk" hreflang="uk" data-title="Десяткова система числення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B9%DB%8C%D8%AF_%D8%A7%D8%B9%D8%B4%D8%A7%D8%B1%DB%8C%DB%81" title="معید اعشاریہ – Urdu" lang="ur" hreflang="ur" data-title="معید اعشاریہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_th%E1%BA%ADp_ph%C3%A2n" title="Hệ thập phân – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ thập phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8D%81%E9%80%B2%E5%88%B6" title="十進制 – Literary Chinese" lang="lzh" hreflang="lzh" 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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">For other uses, see <a href="/wiki/Decimal_(disambiguation)" class="mw-disambig" title="Decimal (disambiguation)">Decimal (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Decimal_digit.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Decimal_digit.png/260px-Decimal_digit.png" decoding="async" width="260" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Decimal_digit.png/390px-Decimal_digit.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Decimal_digit.png/520px-Decimal_digit.png 2x" data-file-width="842" data-file-height="403" /></a><figcaption>Place value of number in decimal system</figcaption></figure> <p>The <b>decimal</b> <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> (also called the <b>base-ten</b> <a href="/wiki/Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral system</a> and <b>denary</b> <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'d' in 'dye'">d</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'n' in 'nigh'">n</span><span title="/ər/: 'er' in 'letter'">ər</span><span title="/i/: 'y' in 'happy'">i</span></span>/</a></span></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or <b>decanary</b>) is the standard system for denoting <a href="/wiki/Integer" title="Integer">integer</a> and non-integer <a href="/wiki/Number" title="Number">numbers</a>. It is the extension to non-integer numbers (<i>decimal fractions</i>) of the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>. The way of denoting numbers in the decimal system is often referred to as <i>decimal notation</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A <b>decimal numeral</b> (also often just <i>decimal</i> or, less correctly, <i>decimal number</i>), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a <a href="/wiki/Decimal_separator" title="Decimal separator">decimal separator</a> (usually "." or "," as in <span class="texhtml">25.9703</span> or <span class="texhtml">3,1415</span>).<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i>Decimal</i> may also refer specifically to the digits after the decimal separator, such as in "<span class="texhtml">3.14</span> is the approximation of <span class="texhtml mvar" style="font-style:italic;">π</span> to <i>two decimals</i>". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. </p><p>The numbers that may be represented in the decimal system are the <a href="#Decimal_fractions"><b>decimal fractions</b></a>. That is, <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractions</a> of the form <span class="texhtml"><i>a</i>/10<sup><i>n</i></sup></span>, where <span class="texhtml"><i>a</i></span> is an integer, and <span class="texhtml"><i>n</i></span> is a <a href="/wiki/Non-negative_integer" class="mw-redirect" title="Non-negative integer">non-negative integer</a>. Decimal fractions also result from the addition of an integer and a <i><a href="/wiki/Fractional_part" title="Fractional part">fractional part</a></i>; the resulting sum sometimes is called a <i>fractional number</i>. </p><p>Decimals are commonly used to <a href="/wiki/Approximation_(mathematics)" class="mw-redirect" title="Approximation (mathematics)">approximate</a> real numbers. By increasing the number of digits after the decimal separator, one can make the <a href="/wiki/Approximation_error" title="Approximation error">approximation errors</a> as small as one wants, when one has a method for computing the new digits. </p><p><span class="anchor" id="terminating_decimal"></span> Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. However, the decimal system has been extended to <i>infinite decimals</i> for representing any <a href="/wiki/Real_number" title="Real number">real number</a>, by using an <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">infinite sequence</a> of digits after the decimal separator (see <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a>). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called <b>terminating decimals</b>. A <i><a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a></i> is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., <span class="texhtml">5.123144144144144... = 5.123<span style="text-decoration:overline;">144</span></span>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> An infinite decimal represents a <a href="/wiki/Rational_number" title="Rational number">rational number</a>, the <a href="/wiki/Quotient" title="Quotient">quotient</a> of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin">Origin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=1" title="Edit section: Origin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Two_hand,_ten_fingers.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Two_hand%2C_ten_fingers.jpg/260px-Two_hand%2C_ten_fingers.jpg" decoding="async" width="260" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Two_hand%2C_ten_fingers.jpg/390px-Two_hand%2C_ten_fingers.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1b/Two_hand%2C_ten_fingers.jpg 2x" data-file-width="400" data-file-height="224" /></a><figcaption>Ten digits on two hands, the possible origin of decimal counting</figcaption></figure> <p>Many <a href="/wiki/Numeral_system" title="Numeral system">numeral systems</a> of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the <a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian numerals</a>, then the <a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi numerals</a>, <a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a>, <a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew numerals</a>, <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a>, and <a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese numerals</a>.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> for representing <a href="/wiki/Integer" title="Integer">integers</a>. This system has been extended to represent some non-integer numbers, called <i><a href="#Decimal_fractions">decimal fractions</a></i> or <i>decimal numbers</i>, for forming the <i>decimal numeral system</i>.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Decimal_notation">Decimal notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=2" title="Edit section: Decimal notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For writing numbers, the decimal system uses ten <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digits</a>, a <a href="/wiki/Decimal_mark" class="mw-redirect" title="Decimal mark">decimal mark</a>, and, for <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>, a <a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a> "−". The decimal digits are <a href="/wiki/0" title="0">0</a>, <a href="/wiki/1" title="1">1</a>, <a href="/wiki/2" title="2">2</a>, <a href="/wiki/3" title="3">3</a>, <a href="/wiki/4" title="4">4</a>, <a href="/wiki/5" title="5">5</a>, <a href="/wiki/6" title="6">6</a>, <a href="/wiki/7" title="7">7</a>, <a href="/wiki/8" title="8">8</a>, <a href="/wiki/9" title="9">9</a>;<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Decimal_separator" title="Decimal separator">decimal separator</a> is the dot "<span class="texhtml">.</span>" in many countries (mostly English-speaking),<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and a comma "<span class="texhtml">,</span>" in other countries.<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For representing a <a href="/wiki/Non-negative_number" class="mw-redirect" title="Non-negative number">non-negative number</a>, a decimal numeral consists of </p> <ul><li>either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}a_{m-1}\ldots a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}a_{m-1}\ldots a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066076aa1595838607cf30164c22d5f9255e65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.692ex; height:2.009ex;" alt="{\displaystyle a_{m}a_{m-1}\ldots a_{0}}"></span></dd></dl></li> <li>or a decimal mark separating two sequences of digits (such as "20.70828")</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c91d178399aef2380bd37147a6100b04f43a115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.543ex; height:2.509ex;" alt="{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}"></span>.</dd></dl></dd></dl> <p>If <span class="texhtml"><i>m</i> > 0</span>, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit <span class="texhtml"><i>a</i><sub><i>m</i></sub></span> is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, <span class="texhtml">3.14 = 03.14 = 003.14</span>. Similarly, if the final digit on the right of the decimal mark is zero—that is, if <span class="texhtml"><i>b</i><sub><i>n</i></sub> = 0</span>—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> for example, <span class="texhtml">15 = 15.0 = 15.00</span> and <span class="texhtml">5.2 = 5.20 = 5.200</span>. </p><p>For representing a <a href="/wiki/Negative_number" title="Negative number">negative number</a>, a minus sign is placed before <span class="texhtml"><i>a</i><sub><i>m</i></sub></span>. </p><p>The numeral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c91d178399aef2380bd37147a6100b04f43a115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.543ex; height:2.509ex;" alt="{\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}"></span> represents the number </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced07f78a8ee96f7c118d78f0508fdd8e236de86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.813ex; height:5.843ex;" alt="{\displaystyle a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}}"></span>.</dd></dl> <p>The <i><a href="/wiki/Integer_part" class="mw-redirect" title="Integer part">integer part</a></i> or <i>integral part</i> of a decimal numeral is the integer written to the left of the decimal separator (see also <a href="/wiki/Truncation" title="Truncation">truncation</a>). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the <i><a href="/wiki/Fractional_part" title="Fractional part">fractional part</a></i>, which equals the difference between the numeral and its integer part. </p><p>When the integral part of a numeral is zero, it may occur, typically in <a href="/wiki/Computing" title="Computing">computing</a>, that the integer part is not written (for example, <span class="texhtml">.1234</span>, instead of <span class="texhtml">0.1234</span>). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation. </p><p>In brief, the contribution of each digit to the value of a number depends on its position in the numeral. 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic</a></li> <li><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul> <hr /> <ul><li><a href="/wiki/Bengali_numerals" title="Bengali numerals">Bengali</a></li> <li><a href="/wiki/Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li> <li><a href="/wiki/Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li> <li><a href="/wiki/Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li> <li><a href="/wiki/Odia_numerals" title="Odia numerals">Odia</a></li> <li><a href="/wiki/Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li> <li><a href="/wiki/Tamil_numerals" title="Tamil numerals">Tamil</a></li> <li><a href="/wiki/Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li> <li><a href="/wiki/Telugu_script#Numerals" title="Telugu script">Telugu</a></li> <li><a href="/wiki/Kannada_script#Numerals" title="Kannada script">Kannada</a></li> <li><a href="/wiki/Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul> <hr /> <ul><li><a href="/wiki/Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li> <li><a href="/wiki/Balinese_numerals" title="Balinese numerals">Balinese</a></li> <li><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></li> <li><a href="/wiki/Javanese_numerals" title="Javanese numerals">Javanese</a></li> <li><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></li> <li><a href="/wiki/Lao_script#Numerals" title="Lao script">Lao</a></li> <li><a href="/wiki/Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li> <li><a href="/wiki/Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li> <li><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a> <ul><li><a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li> <li><a href="/wiki/Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li> <li><a href="/wiki/Japanese_numerals" title="Japanese numerals">Japanese</a></li> <li><a href="/wiki/Korean_numerals" title="Korean numerals">Korean</a></li> <li><a href="/wiki/Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul> <hr /> <dl><dt>Historic</dt></dl> <ul><li><a href="/wiki/Counting_rods" title="Counting rods">Counting rods</a></li> <li><a href="/wiki/Tangut_numerals" title="Tangut numerals">Tangut</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul> <hr /> <dl><dt><a href="/wiki/Ancient_history" title="Ancient history">Ancient</a></dt></dl> <ul><li><a href="/wiki/Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul> <hr /> <dl><dt><a href="/wiki/Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl> <ul><li><a href="/wiki/Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li> <li><a href="/wiki/Maya_numerals" title="Maya numerals">Mayan</a></li> <li><a href="/wiki/Muisca_numerals" title="Muisca numerals">Muisca</a></li> <li><a href="/wiki/Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li> <li><a href="/wiki/Quipu" title="Quipu">Quipu</a></li> <li><a href="/wiki/Rumi_Numeral_Symbols" title="Rumi Numeral Symbols">Rumi</a></li></ul> <hr /> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li> <li><a href="/wiki/Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</a> (Iñupiaq)</li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="/wiki/Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Common radices/bases</dt></dl> <ul><li><a href="/wiki/Binary_number" title="Binary number">2</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">3</a></li> <li><a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">4</a></li> <li><a href="/wiki/Quinary" title="Quinary">5</a></li> <li><a href="/wiki/Senary" title="Senary">6</a></li> <li><a href="/wiki/Octal" title="Octal">8</a></li> <li><a class="mw-selflink selflink">10</a></li> <li><a href="/wiki/Duodecimal" title="Duodecimal">12</a></li> <li><a href="/wiki/Hexadecimal" title="Hexadecimal">16</a></li> <li><a href="/wiki/Vigesimal" title="Vigesimal">20</a></li> <li><a href="/wiki/Sexagesimal" title="Sexagesimal">60</a></li></ul> <hr /> <dl><dt><a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl> <ul><li><a href="/wiki/Bijective_numeration" title="Bijective numeration">Bijective</a><span class="nowrap"> </span>(<a href="/wiki/Unary_numeral_system" title="Unary numeral system">1</a>)</li> <li><a href="/wiki/Signed-digit_representation" title="Signed-digit representation">Signed-digit</a><span class="nowrap"> </span>(<a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li> <li><a href="/wiki/Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<a href="/wiki/Factorial_number_system" title="Factorial number system">factorial</a>)</li> <li><a href="/wiki/Negative_base" title="Negative base">Negative</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex</a><span class="nowrap"> </span>(<a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li> <li><a href="/wiki/Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<a href="/wiki/Golden_ratio_base" title="Golden ratio base">φ</a>)</li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></li></ul></div></div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Non-alphabetic</dt></dl> <ul><li><a href="/wiki/Aegean_numerals" title="Aegean numerals">Aegean</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic</a></li> <li><a href="/wiki/Aztec_script#Numerals" title="Aztec script">Aztec</a></li> <li><a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li> <li><a href="/wiki/Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li> <li><a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li> <li><a href="/wiki/Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li> <li><a href="/wiki/Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li> <li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric counting</a></li> <li><a href="/wiki/Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li> <li><a href="/wiki/Roman_numerals" title="Roman numerals">Roman</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li></ul> <hr /> <dl><dt><a href="/wiki/Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl> <ul><li><a href="/wiki/Abjad_numerals" title="Abjad numerals">Abjad</a></li> <li><a href="/wiki/Armenian_numerals" title="Armenian numerals">Armenian</a></li> <li><a href="/wiki/Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a> <ul><li><a href="/wiki/Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li> <li><a href="/wiki/Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li> <li><a href="/wiki/Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li> <li><a href="/wiki/Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li> <li><a href="/wiki/Georgian_numerals" title="Georgian numerals">Georgian</a></li> <li><a href="/wiki/Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek</a></li> <li><a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numeral_systems" title="Template:Numeral systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numeral_systems" title="Template talk:Numeral systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numeral_systems" title="Special:EditPage/Template:Numeral systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Decimal fractions</b> (sometimes called <b>decimal numbers</b>, especially in contexts involving explicit fractions) are the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> that may be expressed as a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> whose <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> is a <a href="/wiki/Exponentiation" title="Exponentiation">power</a> of ten.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, the decimal expressions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.8,14.89,0.00079,1.618,3.14159}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.8</mn> <mo>,</mo> <mn>14.89</mn> <mo>,</mo> <mn>0.00079</mn> <mo>,</mo> <mn>1.618</mn> <mo>,</mo> <mn>3.14159</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.8,14.89,0.00079,1.618,3.14159}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969d87fbf8f8842dd8fb8d2aa7f4efd97d089c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.944ex; height:2.509ex;" alt="{\displaystyle 0.8,14.89,0.00079,1.618,3.14159}"></span> represent the fractions <span class="texhtml"><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><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span></span>, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1489</span><span class="sr-only">/</span><span class="den">100</span></span>⁠</span></span>, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">79</span><span class="sr-only">/</span><span class="den">100000</span></span>⁠</span></span>, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="sr-only">+</span><span class="tion"><span class="num">809</span><span class="sr-only">/</span><span class="den">500</span></span>⁠</span></span> and <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="sr-only">+</span><span class="tion"><span class="num">314159</span><span class="sr-only">/</span><span class="den">100000</span></span>⁠</span></span>, and therefore denote decimal fractions. An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></span>, 3 not being a power of 10. </p><p>More generally, a decimal with <span class="texhtml"><i>n</i></span> digits after the <a href="/wiki/Decimal_separator" title="Decimal separator">separator</a> (a point or comma) represents the fraction with denominator <span class="texhtml">10<sup><i>n</i></sup></span>, whose numerator is the integer obtained by removing the separator. </p><p>It follows that a number is a decimal fraction <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it has a finite decimal representation. </p><p>Expressed as <a href="/wiki/Fully_reduced_fraction" class="mw-redirect" title="Fully reduced fraction">fully reduced fractions</a>, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mn>2</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mn>4</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mn>5</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>8</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mn>10</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>16</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mn>20</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>25</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e85961582a90be5b08e3607888259eccadfbc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:110.039ex; height:3.009ex;" alt="{\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Approximation_using_decimal_numbers">Approximation using decimal numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=4" title="Edit section: Approximation using decimal numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Decimal numerals do not allow an exact representation for all <a href="/wiki/Real_number" title="Real number">real numbers</a>. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates <span class="texhtml mvar" style="font-style:italic;">π</span>, being less than 10<sup>−5</sup> off; so decimals are widely used in <a href="/wiki/Science" title="Science">science</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a> and everyday life. </p><p>More precisely, for every real number <span class="texhtml mvar" style="font-style:italic;">x</span> and every positive integer <span class="texhtml mvar" style="font-style:italic;">n</span>, there are two decimals <span class="texhtml mvar" style="font-style:italic;"><i>L</i></span> and <span class="texhtml mvar" style="font-style:italic;"><i>u</i></span> with at most <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> digits after the decimal mark such that <span class="texhtml"><i>L</i> ≤ <i>x</i> ≤ <i>u</i></span> and <span class="texhtml">(<i>u</i> − <i>L</i>) = 10<sup>−<i>n</i></sup></span>. </p><p>Numbers are very often obtained as the result of <a href="/wiki/Measurement" title="Measurement">measurement</a>. As measurements are subject to <a href="/wiki/Measurement_uncertainty" title="Measurement uncertainty">measurement uncertainty</a> with a known <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a>, the result of a measurement is well-represented by a decimal with <span class="texhtml"><i>n</i></span> digits after the decimal mark, as soon as the absolute measurement error is bounded from above by <span class="texhtml">10<sup>−<i>n</i></sup></span>. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also <a href="/wiki/Significant_figures" title="Significant figures">significant figures</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Infinite_decimal_expansion">Infinite decimal expansion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=5" title="Edit section: Infinite decimal expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Decimal_representation" title="Decimal representation">Decimal representation</a></div> <p>For a <a href="/wiki/Real_number" title="Real number">real number</a> <span class="texhtml mvar" style="font-style:italic;">x</span> and an integer <span class="texhtml"><i>n</i> ≥ 0</span>, let <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span> denote the (finite) decimal expansion of the greatest number that is not greater than <i><span class="texhtml mvar" style="font-style:italic;">x</span></i> that has exactly <span class="texhtml mvar" style="font-style:italic;">n</span> digits after the decimal mark. Let <span class="texhtml"><i>d</i><sub><i>i</i></sub></span> denote the last digit of <span class="texhtml">[<i>x</i>]<sub><i>i</i></sub></span>. It is straightforward to see that <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span> may be obtained by appending <span class="texhtml"><i>d</i><sub><i>n</i></sub></span> to the right of <span class="texhtml">[<i>x</i>]<sub><i>n</i>−1</sub></span>. This way one has </p> <dl><dd><span class="texhtml">[<i>x</i>]<sub><i>n</i></sub> = [<i>x</i>]<sub>0</sub>.<i>d</i><sub>1</sub><i>d</i><sub>2</sub>...<i>d</i><sub><i>n</i>−1</sub><i>d</i><sub><i>n</i></sub></span>,</dd></dl> <p>and the difference of <span class="texhtml">[<i>x</i>]<sub><i>n</i>−1</sub></span> and <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span> amounts to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mrow> <mo>[</mo> <mi>x</mi> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mo>[</mo> <mi>x</mi> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo><</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa97ac58d939553b57b29422c01a1925a436028e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.966ex; height:3.343ex;" alt="{\displaystyle \left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}}"></span>,</dd></dl> <p>which is either 0, if <span class="texhtml"><i>d</i><sub><i>n</i></sub> = 0</span>, or gets arbitrarily small as <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> tends to infinity. According to the definition of a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>, <i><span class="texhtml mvar" style="font-style:italic;">x</span></i> is the limit of <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span> when <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> tends to <a href="/wiki/Infinity" title="Infinity">infinity</a>. This is written as<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">[</mo> <mi>x</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4726bcc43d70340455e0f0340fa72d52ee7e420d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.295ex; height:2.843ex;" alt="{\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}"></span>or </p> <dl><dd><span class="texhtml"><i>x</i> = [<i>x</i>]<sub>0</sub>.<i>d</i><sub>1</sub><i>d</i><sub>2</sub>...<i>d</i><sub><i>n</i></sub>...</span>,</dd></dl> <p>which is called an <b>infinite decimal expansion</b> of <i><span class="texhtml mvar" style="font-style:italic;">x</span></i>. </p><p>Conversely, for any integer <span class="texhtml">[<i>x</i>]<sub>0</sub></span> and any sequence of digits<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;(d_{n})_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \;(d_{n})_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c8aae98cd94714ebed3607938f206946a7a447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.201ex; height:3.009ex;" alt="{\textstyle \;(d_{n})_{n=1}^{\infty }}"></span> the (infinite) expression <span class="texhtml">[<i>x</i>]<sub>0</sub>.<i>d</i><sub>1</sub><i>d</i><sub>2</sub>...<i>d</i><sub><i>n</i></sub>...</span> is an <i>infinite decimal expansion</i> of a real number <i><span class="texhtml mvar" style="font-style:italic;">x</span></i>. This expansion is unique if neither all <span class="texhtml"><i>d</i><sub><i>n</i></sub></span> are equal to 9 nor all <span class="texhtml"><i>d</i><sub><i>n</i></sub></span> are equal to 0 for <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> large enough (for all <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> greater than some natural number <span class="texhtml mvar" style="font-style:italic;">N</span>). </p><p>If all <span class="texhtml"><i>d</i><sub><i>n</i></sub></span> for <span class="texhtml"><i>n</i> > <i>N</i></span> equal to 9 and <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub> = [<i>x</i>]<sub>0</sub>.<i>d</i><sub>1</sub><i>d</i><sub>2</sub>...<i>d</i><sub><i>n</i></sub></span>, the limit of the sequence<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;([x]_{n})_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>x</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \;([x]_{n})_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ebb69f79147f59b60a7fc1079add44c50c1331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.615ex; height:3.009ex;" alt="{\textstyle \;([x]_{n})_{n=1}^{\infty }}"></span> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: <span class="texhtml"><i>d</i><sub><i>N</i></sub></span>, by <span class="texhtml"><i>d</i><sub><i>N</i></sub> + 1</span>, and replacing all subsequent 9s by 0s (see <a href="/wiki/0.999..." title="0.999...">0.999...</a>). </p><p>Any such decimal fraction, i.e.: <span class="texhtml"><i>d</i><sub><i>n</i></sub> = 0</span> for <span class="texhtml"><i>n</i> > <i>N</i></span>, may be converted to its equivalent infinite decimal expansion by replacing <span class="texhtml"><i>d</i><sub><i>N</i></sub></span> by <span class="texhtml"><i>d</i><sub><i>N</i></sub> − 1</span> and replacing all subsequent 0s by 9s (see <a href="/wiki/0.999..." title="0.999...">0.999...</a>). </p><p>In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span>, and the other containing only 9s after some place, which is obtained by defining <span class="texhtml">[<i>x</i>]<sub><i>n</i></sub></span> as the greatest number that is <i>less</i> than <span class="texhtml mvar" style="font-style:italic;">x</span>, having exactly <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> digits after the decimal mark. </p> <div class="mw-heading mw-heading3"><h3 id="Rational_numbers">Rational numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=6" title="Edit section: Rational numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a></div> <p><a href="/wiki/Long_division" title="Long division">Long division</a> allows computing the infinite decimal expansion of a <a href="/wiki/Rational_number" title="Rational number">rational number</a>. If the rational number is a <a href="#decimal_fraction">decimal fraction</a>, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a <i>repeating decimal</i>. For example, </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">81</span></span>⁠</span> = 0.<span style="white-space: nowrap;"> </span>012345679<span style="white-space: nowrap;"> </span>012... (with the group 012345679 indefinitely repeating).</dd></dl> <p>The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational. </p> <table> <tbody><tr> <td>For example, if <i>x</i> is</td> <td><span class="nowrap">      </span>0.4156156156... </td></tr> <tr> <td>then 10,000<i>x</i> is</td> <td><span class="nowrap">   </span>4156.156156156... </td></tr> <tr> <td>and 10<i>x</i> is</td> <td><span class="nowrap">      </span>4.156156156... </td></tr> <tr> <td>so 10,000<i>x</i> − 10<i>x</i>, i.e. 9,990<i>x</i>, is</td> <td><span class="nowrap">   </span>4152.000000000... </td></tr> <tr> <td>and <i>x</i> is</td> <td><span class="nowrap">   </span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4152</span><span class="sr-only">/</span><span class="den">9990</span></span>⁠</span> </td></tr></tbody></table> <p>or, dividing both numerator and denominator by 6, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">692</span><span class="sr-only">/</span><span class="den">1665</span></span>⁠</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Decimal_computation">Decimal computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=7" title="Edit section: Decimal computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Decimal_multiplication_table.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Decimal_multiplication_table.JPG/300px-Decimal_multiplication_table.JPG" decoding="async" width="300" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Decimal_multiplication_table.JPG/450px-Decimal_multiplication_table.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Decimal_multiplication_table.JPG/600px-Decimal_multiplication_table.JPG 2x" data-file-width="1116" data-file-height="527" /></a><figcaption>Diagram of the world's earliest known multiplication table (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 305 BCE</span>) from the <a href="/wiki/Warring_States_period" title="Warring States period">Warring States period</a></figcaption></figure> <p>Most modern <a href="/wiki/Computer" title="Computer">computer</a> hardware and software systems commonly use a <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary representation</a> internally (although many early computers, such as the <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a> or the <a href="/wiki/IBM_650" title="IBM 650">IBM 650</a>, used decimal representation internally).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For external use by computer specialists, this binary representation is sometimes presented in the related <a href="/wiki/Octal" title="Octal">octal</a> or <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> systems. </p><p>For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.) </p><p>Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of <a href="/wiki/Binary-coded_decimal" title="Binary-coded decimal">binary-coded decimal</a>,<sup id="cite_ref-Schmid_1983_11-0" class="reference"><a href="#cite_note-Schmid_1983-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schmid_1974_12-0" class="reference"><a href="#cite_note-Schmid_1974-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> especially in database implementations, but there are other decimal representations in use (including <a href="/wiki/Decimal_floating_point" title="Decimal floating point">decimal floating point</a> such as in newer revisions of the <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754 Standard for Floating-Point Arithmetic</a>).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span> have no finite binary fractional representation; and is generally impossible for multiplication (or division).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision arithmetic</a> for exact calculations. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=8" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Qinghuajian,_Suan_Biao.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/170px-Qinghuajian%2C_Suan_Biao.jpg" decoding="async" width="170" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/255px-Qinghuajian%2C_Suan_Biao.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/340px-Qinghuajian%2C_Suan_Biao.jpg 2x" data-file-width="1528" data-file-height="2096" /></a><figcaption>The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during the <a href="/wiki/Warring_States" class="mw-redirect" title="Warring States">Warring States</a> period in China.</figcaption></figure> <p>Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Standardized weights used in the <a href="/wiki/Indus_Valley_Civilisation" title="Indus Valley Civilisation">Indus Valley Civilisation</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 3300–1300 BCE</span>) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the <i>Mohenjo-daro ruler</i> – was divided into ten equal parts.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Egyptian_hieroglyphs" title="Egyptian hieroglyphs">Egyptian hieroglyphs</a>, in evidence since around 3000 BCE, used a purely decimal system,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> as did the <a href="/wiki/Linear_A" title="Linear A">Linear A</a> script (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1800–1450 BCE</span>) of the <a href="/wiki/Minoan_civilization" title="Minoan civilization">Minoans</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Linear_B" title="Linear B">Linear B</a> script (c. 1400–1200 BCE) of the <a href="/wiki/Mycenaean_Greece" title="Mycenaean Greece">Mycenaeans</a>. The <a href="/wiki/%C3%9An%C4%9Btice_culture" title="Únětice culture">Únětice culture</a> in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The number system of <a href="/wiki/Classical_Greece" title="Classical Greece">classical Greece</a> also used powers of ten, including an intermediate base of 5, as did <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a>.<sup id="cite_ref-Greek_numerals_24-0" class="reference"><a href="#cite_note-Greek_numerals-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Notably, the polymath <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (c. 287–212 BCE) invented a decimal positional system in his <a href="/wiki/The_Sand_Reckoner" title="The Sand Reckoner">Sand Reckoner</a> which was based on 10<sup>8</sup>.<sup id="cite_ref-Greek_numerals_24-1" class="reference"><a href="#cite_note-Greek_numerals-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Hittites" title="Hittites">Hittite</a> hieroglyphs (since 15th century BCE) were also strictly decimal.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The world's earliest positional decimal system was the Chinese <a href="/wiki/Rod_calculus" title="Rod calculus">rod calculus</a>.<sup id="cite_ref-Lam_28-0" class="reference"><a href="#cite_note-Lam-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Chounumerals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/280px-Chounumerals.svg.png" decoding="async" width="280" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/420px-Chounumerals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/560px-Chounumerals.svg.png 2x" data-file-width="900" data-file-height="300" /></a><figcaption>The world's earliest positional decimal system<br /> Upper row vertical form<br /> Lower row horizontal form</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="History_of_decimal_fractions">History of decimal fractions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=9" title="Edit section: History of decimal fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rod_fraction.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rod_fraction.jpg/150px-Rod_fraction.jpg" decoding="async" width="150" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rod_fraction.jpg/225px-Rod_fraction.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rod_fraction.jpg/300px-Rod_fraction.jpg 2x" data-file-width="381" data-file-height="221" /></a><figcaption>counting rod decimal fraction 1/7</figcaption></figure> <p>Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.<sup id="cite_ref-jnfractn1_29-0" class="reference"><a href="#cite_note-jnfractn1-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Calculations with decimal fractions of lengths were <a href="/wiki/Rod_calculus#Decimal_fraction" title="Rod calculus">performed using positional counting rods</a>, as described in the 3rd–5th century CE <i><a href="/wiki/Sunzi_Suanjing" title="Sunzi Suanjing">Sunzi Suanjing</a></i>. The 5th century CE mathematician <a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> calculated a 7-digit <a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">approximation of <span class="texhtml mvar" style="font-style:italic;">π</span></a>. <a href="/wiki/Qin_Jiushao" title="Qin Jiushao">Qin Jiushao</a>'s book <i><a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a></i> (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The number 0.96644 is denoted </p> <dl><dd><span title="Chinese-language text"><span lang="zh">寸</span></span></dd> <dd><span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Counting_rod_0.png/18px-Counting_rod_0.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png 1.5x" data-file-width="23" data-file-height="23" /></a></span> <span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_h9_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Counting_rod_h9_num.png/18px-Counting_rod_h9_num.png" decoding="async" width="18" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Counting_rod_h9_num.png/27px-Counting_rod_h9_num.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_h9_num.png 2x" data-file-width="29" data-file-height="31" /></a></span> <span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Counting_rod_v6.png/18px-Counting_rod_v6.png" decoding="async" width="18" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Counting_rod_v6.png/27px-Counting_rod_v6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png 2x" data-file-width="29" data-file-height="32" /></a></span> <span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_h6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Counting_rod_h6.png/18px-Counting_rod_h6.png" decoding="async" width="18" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Counting_rod_h6.png/27px-Counting_rod_h6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c1/Counting_rod_h6.png 2x" data-file-width="29" data-file-height="31" /></a></span> <span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Counting_rod_v4.png/18px-Counting_rod_v4.png" decoding="async" width="18" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png 1.5x" data-file-width="25" data-file-height="29" /></a></span> <span typeof="mw:File/Frameless"><a href="/wiki/File:Counting_rod_h4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Counting_rod_h4.png/18px-Counting_rod_h4.png" decoding="async" width="18" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Counting_rod_h4.png/27px-Counting_rod_h4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/ce/Counting_rod_h4.png 2x" data-file-width="29" data-file-height="25" /></a></span>.</dd></dl> <p>Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.<sup id="cite_ref-Lam_28-1" class="reference"><a href="#cite_note-Lam-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a> introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.<sup id="cite_ref-Lam_28-2" class="reference"><a href="#cite_note-Lam-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>Positional decimal fractions appear for the first time in a book by the Arab mathematician <a href="/wiki/Abu%27l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a> written in the 10th century.<sup id="cite_ref-Berggren_32-0" class="reference"><a href="#cite_note-Berggren-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The Jewish mathematician <a href="/wiki/Immanuel_Bonfils" title="Immanuel Bonfils">Immanuel Bonfils</a> used decimal fractions around 1350 but did not develop any notation to represent them.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The Persian mathematician <a href="/wiki/Jamshid_al-Kashi" title="Jamshid al-Kashi">Jamshid al-Kashi</a> used, and claimed to have discovered, decimal fractions in the 15th century.<sup id="cite_ref-Berggren_32-1" class="reference"><a href="#cite_note-Berggren-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div style="float: right;"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Stevin-decimal_notation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Stevin-decimal_notation.svg/540px-Stevin-decimal_notation.svg.png" decoding="async" width="540" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Stevin-decimal_notation.svg/810px-Stevin-decimal_notation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Stevin-decimal_notation.svg/1080px-Stevin-decimal_notation.svg.png 2x" data-file-width="540" data-file-height="90" /></a></span></div> <p>A forerunner of modern European decimal notation was introduced by <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> in the 16th century. Stevin's influential booklet <i><a href="/wiki/De_Thiende" title="De Thiende">De Thiende</a></i> ("the art of tenths") was first published in Dutch in 1585 and translated into French as <i>La Disme</i>.<sup id="cite_ref-van_34-0" class="reference"><a href="#cite_note-van-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/John_Napier" title="John Napier">John Napier</a> introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.<sup id="cite_ref-constructionIA_35-0" class="reference"><a href="#cite_note-constructionIA-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 8, archive p. 32)">: p. 8, archive p. 32) </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Natural_languages">Natural languages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=10" title="Edit section: Natural languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A method of expressing every possible <a href="/wiki/Natural_number" title="Natural number">natural number</a> using a set of ten symbols emerged in India.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Several Indian languages show a straightforward decimal system. <a href="/wiki/Dravidian_languages" title="Dravidian languages">Dravidian languages</a> have numbers between 10 and 20 expressed in a regular pattern of addition to 10.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Hungarian_language" title="Hungarian language">Hungarian language</a> also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty"). </p><p>A straightforward decimal rank system with a word for each order (10 <span title="Chinese-language text"><span lang="zh">十</span></span>, 100 <span title="Chinese-language text"><span lang="zh">百</span></span>, 1000 <span title="Chinese-language text"><span lang="zh">千</span></span>, 10,000 <span title="Chinese-language text"><span lang="zh">万</span></span>), and in which 11 is expressed as <i>ten-one</i> and 23 as <i>two-ten-three</i>, and 89,345 is expressed as 8 (ten thousands) <span title="Chinese-language text"><span lang="zh">万</span></span> 9 (thousand) <span title="Chinese-language text"><span lang="zh">千</span></span> 3 (hundred) <span title="Chinese-language text"><span lang="zh">百</span></span> 4 (tens) <span title="Chinese-language text"><span lang="zh">十</span></span> 5 is found in <a href="/wiki/Chinese_language" title="Chinese language">Chinese</a>, and in <a href="/wiki/Vietnamese_language" title="Vietnamese language">Vietnamese</a> with a few irregularities. <a href="/wiki/Japanese_language" title="Japanese language">Japanese</a>, <a href="/wiki/Korean_language" title="Korean language">Korean</a>, and <a href="/wiki/Thai_language" title="Thai language">Thai</a> have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen". </p><p>Incan languages such as <a href="/wiki/Quechuan_languages" title="Quechuan languages">Quechua</a> and <a href="/wiki/Aymara_language" title="Aymara language">Aymara</a> have an almost straightforward decimal system, in which 11 is expressed as <i>ten with one</i> and 23 as <i>two-ten with three</i>. </p><p>Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_bases">Other bases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=11" title="Edit section: Other bases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar nomobile nowraplinks plainlist" style="width:auto"><tbody><tr><td class="sidebar-above" style="background:lavender; font-size:120%;"> <a href="/wiki/Unit_of_measurement" title="Unit of measurement">Units</a> of<br /><a href="/wiki/Units_of_information" title="Units of information">information</a></td></tr><tr><th class="sidebar-heading"> <a href="/wiki/Information_theory" title="Information theory">Information-theoretic</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Shannon_(unit)" title="Shannon (unit)">shannon</a> (<a href="/wiki/Binary_number" title="Binary number">base 2</a>)</li> <li><a href="/wiki/Nat_(unit)" title="Nat (unit)">nat</a> (<a href="/wiki/Natural_logarithm" title="Natural logarithm">base e</a>)</li> <li><a href="/wiki/Hartley_(unit)" title="Hartley (unit)">hartley</a> (<a class="mw-selflink selflink">base 10</a>)</li></ul></td> </tr><tr><th class="sidebar-heading"> Data storage</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bit" title="Bit">bit</a> (<a href="/wiki/Binary_number" title="Binary number">binary</a>)</li> <li>trit (<a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">ternary</a>)</li> <li><a href="/wiki/Hartley_(unit)" title="Hartley (unit)">dit</a> (<a class="mw-selflink selflink">decimal</a>)</li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Qubit" title="Qubit">qubit</a> (binary)</li> <li><a href="/wiki/Qutrit" title="Qutrit">qutrit</a> (ternary)</li> <li><a href="/wiki/Qudit" class="mw-redirect" title="Qudit">qudit</a> (<i>d</i>-dimensional)</li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Fundamental_info_units" title="Template:Fundamental info units"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fundamental_info_units" title="Template talk:Fundamental info units"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fundamental_info_units" title="Special:EditPage/Template:Fundamental info units"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Some cultures do, or did, use other bases of numbers. </p> <ul><li><a href="/wiki/Pre-Columbian" class="mw-redirect" title="Pre-Columbian">Pre-Columbian</a> <a href="/wiki/Mesoamerica" title="Mesoamerica">Mesoamerican</a> cultures such as the <a href="/wiki/Maya_numerals" title="Maya numerals">Maya</a> used a <a href="/wiki/Vigesimal" title="Vigesimal">base-20</a> system (perhaps based on using all twenty fingers and <a href="/wiki/Toe" title="Toe">toes</a>).</li> <li>The <a href="/wiki/Yuki_tribe" class="mw-redirect" title="Yuki tribe">Yuki</a> language in <a href="/wiki/California" title="California">California</a> and the Pamean languages<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Mexico" title="Mexico">Mexico</a> have <a href="/wiki/Octal" title="Octal">octal</a> (<a href="/wiki/Radix" title="Radix">base</a>-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup></li> <li>The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> Where this counting system is known, it is based on the "<a href="/wiki/Long_hundred" title="Long hundred">long hundred</a>" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's <a rel="nofollow" class="external text" href="https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon">Introduction to Old Norse</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160415205641/https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon">Archived</a> 2016-04-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. <a rel="nofollow" class="external text" href="http://ads.ahds.ac.uk/catalogue/adsdata/arch-352-1/dissemination/pdf/vol_123/123_395_418.pdf">Goodare</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged January 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup> details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></li> <li>Many or all of the <a href="/wiki/Chumashan_languages" title="Chumashan languages">Chumashan languages</a> originally used a <a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">base-4</a> counting system, in which the names for numbers were structured according to multiples of 4 and <a href="/wiki/Hexadecimal" title="Hexadecimal">16</a>.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></li> <li>Many languages<sup id="cite_ref-Hammarstrom_2010_46-0" class="reference"><a href="#cite_note-Hammarstrom_2010-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> use <a href="/wiki/Quinary" title="Quinary">quinary (base-5)</a> number systems, including <a href="/wiki/Gumatj_language" class="mw-redirect" title="Gumatj language">Gumatj</a>, <a href="/wiki/Nunggubuyu_language" title="Nunggubuyu language">Nunggubuyu</a>,<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Kuurn_Kopan_Noot_language" class="mw-redirect" title="Kuurn Kopan Noot language">Kuurn Kopan Noot</a><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Saraveca" class="mw-redirect" title="Saraveca">Saraveca</a>. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.</li> <li>Some <a href="/wiki/Nigeria" title="Nigeria">Nigerians</a> use <a href="/wiki/Duodecimal" title="Duodecimal">duodecimal</a> systems.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> So did some small communities in India and Nepal, as indicated by their languages.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Huli_language" title="Huli language">Huli language</a> of <a href="/wiki/Papua_New_Guinea" title="Papua New Guinea">Papua New Guinea</a> is reported to have <a href="/wiki/Pentadecimal" class="mw-redirect" title="Pentadecimal">base-15</a> numbers.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> <i>Ngui</i> means 15, <i>ngui ki</i> means 15 × 2 = 30, and <i>ngui ngui</i> means 15 × 15 = 225.</li> <li><a href="/wiki/Umbu-Ungu_language" class="mw-redirect" title="Umbu-Ungu language">Umbu-Ungu</a>, also known as Kakoli, is reported to have <a href="/wiki/Base_24" class="mw-redirect" title="Base 24">base-24</a> numbers.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> <i>Tokapu</i> means 24, <i>tokapu talu</i> means 24 × 2 = 48, and <i>tokapu tokapu</i> means 24 × 24 = 576.</li> <li><a href="/wiki/Ngiti_language" title="Ngiti language">Ngiti</a> is reported to have a <a href="/wiki/Base_32" class="mw-redirect" title="Base 32">base-32</a> number system with base-4 cycles.<sup id="cite_ref-Hammarstrom_2010_46-1" class="reference"><a href="#cite_note-Hammarstrom_2010-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Ndom_language" title="Ndom language">Ndom language</a> of <a href="/wiki/Papua_New_Guinea" title="Papua New Guinea">Papua New Guinea</a> is reported to have <a href="/wiki/Base-6" class="mw-redirect" title="Base-6">base-6</a> numerals.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> <i>Mer</i> means 6, <i>mer an thef</i> means 6 × 2 = 12, <i>nif</i> means 36, and <i>nif thef</i> means 36×2 = 72.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Algorism" title="Algorism">Algorism</a></li> <li><a href="/wiki/Binary-coded_decimal" title="Binary-coded decimal">Binary-coded decimal</a> (BCD)</li> <li><a href="/wiki/Decimal_classification" title="Decimal classification">Decimal classification</a></li> <li><a href="/wiki/Decimal_computer" title="Decimal computer">Decimal computer</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal time</a></li> <li><a href="/wiki/Decimal_representation" title="Decimal representation">Decimal representation</a></li> <li><a href="/wiki/Decimal_section_numbering" class="mw-redirect" title="Decimal section numbering">Decimal section numbering</a></li> <li><a href="/wiki/Decimal_separator" title="Decimal separator">Decimal separator</a></li> <li><a href="/wiki/Decimalisation" title="Decimalisation">Decimalisation</a></li> <li><a href="/wiki/Densely_packed_decimal" title="Densely packed decimal">Densely packed decimal</a> (DPD)</li> <li><a href="/wiki/Duodecimal" title="Duodecimal">Duodecimal</a></li> <li><a href="/wiki/Octal" title="Octal">Octal</a></li> <li><a href="/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li> <li><a href="/wiki/Serial_decimal" title="Serial decimal">Serial decimal</a></li> <li><a href="/wiki/Metric_prefix" title="Metric prefix">Metric prefix</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=13" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Sometimes, the extra zeros are used for indicating the <a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">accuracy</a> of a measurement. For example, "15.00 m" may indicate that the measurement error is less than one centimetre (0.01 m), while "15 m" may mean that the length is roughly fifteen metres and that the error may exceed 10 centimetres.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decimal&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFReference-OED-denary" class="citation encyclopaedia cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=denary">"denary"</a></span>. <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> (Online ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=denary&rft.btitle=Oxford+English+Dictionary&rft.edition=Online&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fwww.oed.com%2Fsearch%2Fdictionary%2F%3Fq%3Ddenary&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYongSe2004" class="citation book cs1">Yong, Lam Lay; Se, Ang Tian (April 2004). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1142/5425"><i>Fleeting Footsteps</i></a>. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. 268. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F5425">10.1142/5425</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-696-0" title="Special:BookSources/978-981-238-696-0"><bdi>978-981-238-696-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401132256/https://www.worldscientific.com/worldscibooks/10.1142/5425">Archived</a> from the original on April 1, 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">March 17,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fleeting+Footsteps&rft.pages=268&rft.pub=World+Scientific&rft.date=2004-04&rft_id=info%3Adoi%2F10.1142%2F5425&rft.isbn=978-981-238-696-0&rft.aulast=Yong&rft.aufirst=Lam+Lay&rft.au=Se%2C+Ang+Tian&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1142%2F5425&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein2022" class="citation web cs1">Weisstein, Eric W. (March 10, 2022). <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DecimalPoint.html">"Decimal Point"</a>. <i>Wolfram MathWorld</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220321195047/https://mathworld.wolfram.com/DecimalPoint.html">Archived</a> from the original on March 21, 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">March 17,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Decimal+Point&rft.date=2022-03-10&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDecimalPoint.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">The <a href="/wiki/Vinculum_(symbol)" title="Vinculum (symbol)">vinculum (overline)</a> in 5.123<span style="text-decoration: overline;">144</span> indicates that the '144' sequence repeats indefinitely, i.e. <span class="nowrap"><span data-sort-value="7000512314414414414♠"></span>5.123<span style="margin-left:.25em;">144</span><span style="margin-left:.25em;">144</span><span style="margin-left:.25em;">144</span><span style="margin-left:.25em;">144</span>...</span>.</span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLockhart2017" class="citation book cs1">Lockhart, Paul (2017). <i>Arithmetic</i>. Cambridge, Massachusetts London, England: The Belknap Press of Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-97223-0" title="Special:BookSources/978-0-674-97223-0"><bdi>978-0-674-97223-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetic&rft.place=Cambridge%2C+Massachusetts+London%2C+England&rft.pub=The+Belknap+Press+of+Harvard+University+Press&rft.date=2017&rft.isbn=978-0-674-97223-0&rft.aulast=Lockhart&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">In some countries, such as <a href="/wiki/Arabic" title="Arabic">Arabic</a>-speaking ones, other <a href="/wiki/Glyph" title="Glyph">glyphs</a> are used for the digits</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Decimal.html">"Decimal"</a>. <i>mathworld.wolfram.com</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200318204545/https://mathworld.wolfram.com/Decimal.html">Archived</a> from the original on 2020-03-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Decimal&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDecimal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Decimal_fraction">"Decimal Fraction"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131211035917/http://www.encyclopediaofmath.org/index.php/Decimal_fraction">Archived</a> from the original on 2013-12-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-06-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Decimal+Fraction&rft.btitle=Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FDecimal_fraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">"Fingers or Fists? (The Choice of Decimal or Binary Representation)", <a href="/wiki/Werner_Buchholz" title="Werner Buchholz">Werner Buchholz</a>, <i>Communications of the ACM</i>, Vol. 2 #12, pp. 3–11, ACM Press, December 1959.</span> </li> <li id="cite_note-Schmid_1983-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schmid_1983_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmid1983" class="citation book cs1"><a href="/wiki/Hermann_Schmid_(computer_scientist)" class="mw-redirect" title="Hermann Schmid (computer scientist)">Schmid, Hermann</a> (1983) [1974]. <i>Decimal Computation</i> (1 (reprint) ed.). Malabar, Florida: Robert E. Krieger Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89874-318-4" title="Special:BookSources/0-89874-318-4"><bdi>0-89874-318-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decimal+Computation&rft.place=Malabar%2C+Florida&rft.edition=1+%28reprint%29&rft.pub=Robert+E.+Krieger+Publishing+Company&rft.date=1983&rft.isbn=0-89874-318-4&rft.aulast=Schmid&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-Schmid_1974-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schmid_1974_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmid1974" class="citation book cs1"><a href="/wiki/Hermann_Schmid_(computer_scientist)" class="mw-redirect" title="Hermann Schmid (computer scientist)">Schmid, Hermann</a> (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/decimalcomputati0000schm"><i>Decimal Computation</i></a></span> (1st ed.). Binghamton, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-76180-X" title="Special:BookSources/0-471-76180-X"><bdi>0-471-76180-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decimal+Computation&rft.place=Binghamton%2C+New+York&rft.edition=1st&rft.pub=John+Wiley+%26+Sons&rft.date=1974&rft.isbn=0-471-76180-X&rft.aulast=Schmid&rft.aufirst=Hermann&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdecimalcomputati0000schm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><i>Decimal Floating-Point: Algorism for Computers</i>, <a href="/wiki/Mike_Cowlishaw" title="Mike Cowlishaw">Cowlishaw, Mike F.</a>, Proceedings <a href="/wiki/16th_IEEE_Symposium_on_Computer_Arithmetic" class="mw-redirect" title="16th IEEE Symposium on Computer Arithmetic">16th IEEE Symposium on Computer Arithmetic</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7695-1894-X" title="Special:BookSources/0-7695-1894-X">0-7695-1894-X</a>, pp. 104–11, IEEE Comp. Soc., 2003</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://speleotrove.com/decimal/decifaq.html">"Decimal Arithmetic – FAQ"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090429044324/http://speleotrove.com/decimal/decifaq.html">Archived</a> from the original on 2009-04-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Decimal+Arithmetic+%E2%80%93+FAQ&rft_id=http%3A%2F%2Fspeleotrove.com%2Fdecimal%2Fdecifaq.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.dec.usc.es/arith16/papers/paper-107.pdf">Decimal Floating-Point: Algorism for Computers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20031116234555/http://www.dec.usc.es/arith16/papers/paper-107.pdf">Archived</a> 2003-11-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/Cowlishaw" title="Cowlishaw">Cowlishaw</a>, M. F., <i>Proceedings <a href="/wiki/16th_IEEE_Symposium_on_Computer_Arithmetic" class="mw-redirect" title="16th IEEE Symposium on Computer Arithmetic">16th IEEE Symposium on Computer Arithmetic</a></i> (<a rel="nofollow" class="external text" href="http://www.dec.usc.es/arith16/">ARITH 16</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100819005155/http://www.dec.usc.es/arith16/">Archived</a> 2010-08-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7695-1894-X" title="Special:BookSources/0-7695-1894-X">0-7695-1894-X</a>, pp. 104–11, IEEE Comp. Soc., June 2003</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDantzig1954" class="citation cs2">Dantzig, Tobias (1954), <i>Number / The Language of Science</i> (4th ed.), The Free Press (Macmillan Publishing Co.), p. 12, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-02-906990-4" title="Special:BookSources/0-02-906990-4"><bdi>0-02-906990-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+%2F+The+Language+of+Science&rft.pages=12&rft.edition=4th&rft.pub=The+Free+Press+%28Macmillan+Publishing+Co.%29&rft.date=1954&rft.isbn=0-02-906990-4&rft.aulast=Dantzig&rft.aufirst=Tobias&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Sergent, Bernard (1997), <i>Genèse de l'Inde</i> (in French), Paris: Payot, p. 113, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/2-228-89116-9" title="Special:BookSources/2-228-89116-9">2-228-89116-9</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoppa2006" class="citation journal cs1">Coppa, A.; et al. (2006). "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population". <i>Nature</i>. <b>440</b> (7085): 755–56. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Natur.440..755C">2006Natur.440..755C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F440755a">10.1038/440755a</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16598247">16598247</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:6787162">6787162</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Early+Neolithic+tradition+of+dentistry%3A+Flint+tips+were+surprisingly+effective+for+drilling+tooth+enamel+in+a+prehistoric+population&rft.volume=440&rft.issue=7085&rft.pages=755-56&rft.date=2006&rft_id=info%3Adoi%2F10.1038%2F440755a&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6787162%23id-name%3DS2CID&rft_id=info%3Apmid%2F16598247&rft_id=info%3Abibcode%2F2006Natur.440..755C&rft.aulast=Coppa&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan <i>Civilisation: A Contemporary Perspective</i>, New Delhi: Oxford and IBH Publishing Co., pp. 113–24</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Georges Ifrah: <i>From One to Zero. A Universal History of Numbers</i>, Penguin Books, 1988, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-009919-0" title="Special:BookSources/0-14-009919-0">0-14-009919-0</a>, pp. 200–13 (Egyptian Numerals)</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-42165-0" title="Special:BookSources/978-0-486-42165-0">978-0-486-42165-0</a>, p. 50</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Georges Ifrah: <i>From One to Zero. A Universal History of Numbers</i>, Penguin Books, 1988, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-009919-0" title="Special:BookSources/0-14-009919-0">0-14-009919-0</a>, pp. 213–18 (Cretan numerals)</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrauseKutscher2017" class="citation book cs1">Krause, Harald; Kutscher, Sabrina (2017). "Spangenbarrenhort Oberding: Zusammenfassung und Ausblick". <a rel="nofollow" class="external text" href="https://www.academia.edu/34550316"><i>Spangenbarrenhort Oberding</i></a>. Museum Erding. pp. 238–243. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-9817606-5-1" title="Special:BookSources/978-3-9817606-5-1"><bdi>978-3-9817606-5-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Spangenbarrenhort+Oberding%3A+Zusammenfassung+und+Ausblick&rft.btitle=Spangenbarrenhort+Oberding&rft.pages=238-243&rft.pub=Museum+Erding&rft.date=2017&rft.isbn=978-3-9817606-5-1&rft.aulast=Krause&rft.aufirst=Harald&rft.au=Kutscher%2C+Sabrina&rft_id=https%3A%2F%2Fwww.academia.edu%2F34550316&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-Greek_numerals-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Greek_numerals_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Greek_numerals_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html">"Greek numbers"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190721085640/http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html">Archived</a> from the original on 2019-07-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-07-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Greek+numbers&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FHistTopics%2FGreek_numbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="/wiki/Karl_Menninger_(mathematics)" title="Karl Menninger (mathematics)">Menninger, Karl</a>: <i>Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl</i>, Vandenhoeck und Ruprecht, 3rd. ed., 1979, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-525-40725-4" title="Special:BookSources/3-525-40725-4">3-525-40725-4</a>, pp. 150–53</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Georges Ifrah: <i>From One to Zero. A Universal History of Numbers</i>, Penguin Books, 1988, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-009919-0" title="Special:BookSources/0-14-009919-0">0-14-009919-0</a>, pp. 218f. (The Hittite hieroglyphic system)</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a href="/wiki/Lam_Lay_Yong" title="Lam Lay Yong">Lam Lay Yong</a> et al. The Fleeting Footsteps pp. 137–39</span> </li> <li id="cite_note-Lam-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lam_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lam_28-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lam_28-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Lam_Lay_Yong" title="Lam Lay Yong">Lam Lay Yong</a>, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", <i>Chinese Science</i>, 1996 p. 38, Kurt Vogel notation</span> </li> <li id="cite_note-jnfractn1-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-jnfractn1_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_Needham1959" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Joseph_Needham" title="Joseph Needham">Joseph Needham</a> (1959). "19.2 Decimals, Metrology, and the Handling of Large Numbers". <a href="/wiki/Science_and_Civilisation_in_China" title="Science and Civilisation in China"><i>Science and Civilisation in China</i></a>. Vol. III, "Mathematics and the Sciences of the Heavens and the Earth". Cambridge University Press. pp. 82–90.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=19.2+Decimals%2C+Metrology%2C+and+the+Handling+of+Large+Numbers&rft.btitle=Science+and+Civilisation+in+China&rft.pages=82-90&rft.pub=Cambridge+University+Press&rft.date=1959&rft.au=Joseph+Needham&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-33782-2" title="Special:BookSources/3-540-33782-2">3-540-33782-2</a></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay_Yong" class="citation journal cs1"><a href="/wiki/Lam_Lay_Yong" title="Lam Lay Yong">Lay Yong, Lam</a>. "A Chinese Genesis, Rewriting the history of our numeral system". <i>Archive for History of Exact Sciences</i>. <b>38</b>: 101–08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=A+Chinese+Genesis%2C+Rewriting+the+history+of+our+numeral+system&rft.volume=38&rft.pages=101-08&rft.aulast=Lay+Yong&rft.aufirst=Lam&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-Berggren-32"><span class="mw-cite-backlink">^ <a href="#cite_ref-Berggren_32-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Berggren_32-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerggren2007" class="citation book cs1">Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.). <i>The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook</i>. Princeton University Press. p. 530. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+in+Medieval+Islam&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India%2C+and+Islam%3A+A+Sourcebook&rft.pages=530&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rft.aulast=Berggren&rft.aufirst=J.+Lennart&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><a href="/wiki/Solomon_Gandz" title="Solomon Gandz">Gandz, S.</a>: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.</span> </li> <li id="cite_note-van-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-van_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._L._van_der_Waerden1985" class="citation book cs1"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">B. L. van der Waerden</a> (1985). <i>A History of Algebra. From Khwarizmi to Emmy Noether</i>. Berlin: Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Algebra.+From+Khwarizmi+to+Emmy+Noether&rft.place=Berlin&rft.pub=Springer-Verlag&rft.date=1985&rft.au=B.+L.+van+der+Waerden&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-constructionIA-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-constructionIA_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNapier1889" class="citation book cs1">Napier, John (1889) [1620]. <i><a href="https://commons.wikimedia.org/wiki/File:The_Construction_of_the_Wonderful_Canon_of_Logarithms.djvu" class="extiw" title="commons:File:The Construction of the Wonderful Canon of Logarithms.djvu">The Construction of the Wonderful Canon of Logarithms</a></i>. Translated by Macdonald, William Rae. Edinburgh: Blackwood & Sons – via Internet Archive. <q>In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Construction+of+the+Wonderful+Canon+of+Logarithms&rft.place=Edinburgh&rft.pub=Blackwood+%26+Sons&rft.date=1889&rft.aulast=Napier&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/">"Indian numerals"</a>. <i>Ancient Indian mathematics</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ancient+Indian+mathematics&rft.atitle=Indian+numerals&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FIndian_numerals%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a class="external text" href="https://en.wiktionary.org/wiki/Appendix:Cognate_sets_for_Dravidian_languages">"Appendix:Cognate sets for Dravidian languages"</a>, <i>Wiktionary, the free dictionary</i>, 2024-09-25<span class="reference-accessdate">, retrieved <span class="nowrap">2024-11-09</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wiktionary%2C+the+free+dictionary&rft.atitle=Appendix%3ACognate+sets+for+Dravidian+languages&rft.date=2024-09-25&rft_id=https%3A%2F%2Fen.wiktionary.org%2Fwiki%2FAppendix%3ACognate_sets_for_Dravidian_languages&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAzar1999" class="citation journal cs1">Azar, Beth (1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071021015527/http://www.apa.org/monitor/apr99/english.html">"English words may hinder math skills development"</a>. <i>American Psychological Association Monitor</i>. <b>30</b> (4). Archived from <a rel="nofollow" class="external text" href="http://www.apa.org/monitor/apr99/english.html">the original</a> on 2007-10-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Psychological+Association+Monitor&rft.atitle=English+words+may+hinder+math+skills+development&rft.volume=30&rft.issue=4&rft.date=1999&rft.aulast=Azar&rft.aufirst=Beth&rft_id=http%3A%2F%2Fwww.apa.org%2Fmonitor%2Fapr99%2Fenglish.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvelino2006" class="citation journal cs1">Avelino, Heriberto (2006). <a rel="nofollow" class="external text" href="http://linguistics.berkeley.edu/~avelino/Avelino_2006.pdf">"The typology of Pame number systems and the limits of Mesoamerica as a linguistic area"</a> <span class="cs1-format">(PDF)</span>. <i>Linguistic Typology</i>. <b>10</b> (1): 41–60. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2FLINGTY.2006.002">10.1515/LINGTY.2006.002</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:20412558">20412558</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060712201924/http://www.linguistics.berkeley.edu/~avelino/Avelino_2006.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2006-07-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linguistic+Typology&rft.atitle=The+typology+of+Pame+number+systems+and+the+limits+of+Mesoamerica+as+a+linguistic+area&rft.volume=10&rft.issue=1&rft.pages=41-60&rft.date=2006&rft_id=info%3Adoi%2F10.1515%2FLINGTY.2006.002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A20412558%23id-name%3DS2CID&rft.aulast=Avelino&rft.aufirst=Heriberto&rft_id=http%3A%2F%2Flinguistics.berkeley.edu%2F~avelino%2FAvelino_2006.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarcia_Ascher" class="citation news cs1"><a href="/wiki/Marcia_Ascher" title="Marcia Ascher">Marcia Ascher</a>. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal. <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/2686959">2686959</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Ethnomathematics%3A+A+Multicultural+View+of+Mathematical+Ideas&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2686959%23id-name%3DJSTOR&rft.au=Marcia+Ascher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcClean1958" class="citation cs2">McClean, R. J. (July 1958), "Observations on the Germanic numerals", <i>German Life and Letters</i>, <b>11</b> (4): 293–99, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1468-0483.1958.tb00018.x">10.1111/j.1468-0483.1958.tb00018.x</a>, <q>Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=German+Life+and+Letters&rft.atitle=Observations+on+the+Germanic+numerals&rft.volume=11&rft.issue=4&rft.pages=293-99&rft.date=1958-07&rft_id=info%3Adoi%2F10.1111%2Fj.1468-0483.1958.tb00018.x&rft.aulast=McClean&rft.aufirst=R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span>.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoyles1987" class="citation cs2">Voyles, Joseph (October 1987), "The cardinal numerals in pre-and proto-Germanic", <i>The Journal of English and Germanic Philology</i>, <b>86</b> (4): 487–95, <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/27709904">27709904</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+English+and+Germanic+Philology&rft.atitle=The+cardinal+numerals+in+pre-and+proto-Germanic&rft.volume=86&rft.issue=4&rft.pages=487-95&rft.date=1987-10&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27709904%23id-name%3DJSTOR&rft.aulast=Voyles&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span>.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStevenson1890" class="citation journal cs1 cs1-prop-long-vol">Stevenson, W.H. (1890). "The Long Hundred and its uses in England". <i>Archaeological Review</i>. December 1889: 313–22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archaeological+Review&rft.atitle=The+Long+Hundred+and+its+uses+in+England&rft.volume=December+1889&rft.pages=313-22&rft.date=1890&rft.aulast=Stevenson&rft.aufirst=W.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoole,_Reginald_Lane2006" class="citation book cs1">Poole, Reginald Lane (2006). <i>The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911</i>. Clark, NJ: Lawbook Exchange. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58477-658-7" title="Special:BookSources/1-58477-658-7"><bdi>1-58477-658-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/76960942">76960942</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Exchequer+in+the+twelfth+century+%3A+the+Ford+lectures+delivered+in+the+University+of+Oxford+in+Michaelmas+term%2C+1911&rft.place=Clark%2C+NJ&rft.pub=Lawbook+Exchange&rft.date=2006&rft_id=info%3Aoclcnum%2F76960942&rft.isbn=1-58477-658-7&rft.au=Poole%2C+Reginald+Lane&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">There is a surviving list of <a href="/wiki/Venture%C3%B1o_language" title="Ventureño language">Ventureño language</a> number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in <i>Native American Mathematics</i>, edited by Michael P. Closs (1986), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-292-75531-7" title="Special:BookSources/0-292-75531-7">0-292-75531-7</a>.</span> </li> <li id="cite_note-Hammarstrom_2010-46"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hammarstrom_2010_46-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hammarstrom_2010_46-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHammarström2007" class="citation book cs1">Hammarström, Harald (17 May 2007). "Rarities in Numeral Systems". In Wohlgemuth, Jan; Cysouw, Michael (eds.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070819214057/http://www.cs.chalmers.se/~harald2/rarapaper.pdf"><i>Rethinking Universals: How rarities affect linguistic theory</i></a> <span class="cs1-format">(PDF)</span>. Empirical Approaches to Language Typology. Vol. 45. Berlin: Mouton de Gruyter (published 2010). Archived from <a rel="nofollow" class="external text" href="http://www.cs.chalmers.se/~harald2/rarapaper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 19 August 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rarities+in+Numeral+Systems&rft.btitle=Rethinking+Universals%3A+How+rarities+affect+linguistic+theory&rft.place=Berlin&rft.series=Empirical+Approaches+to+Language+Typology&rft.pub=Mouton+de+Gruyter&rft.date=2007-05-17&rft.aulast=Hammarstr%C3%B6m&rft.aufirst=Harald&rft_id=http%3A%2F%2Fwww.cs.chalmers.se%2F~harald2%2Frarapaper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris1982" class="citation journal cs1">Harris, John (1982). Hargrave, Susanne (ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070831202737/http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf">"Facts and fallacies of aboriginal number systems"</a> <span class="cs1-format">(PDF)</span>. <i>Work Papers of SIL-AAB Series B</i>. <b>8</b>: 153–81. Archived from <a rel="nofollow" class="external text" href="https://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0029743_v_a.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2007-08-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Work+Papers+of+SIL-AAB+Series+B&rft.atitle=Facts+and+fallacies+of+aboriginal+number+systems&rft.volume=8&rft.pages=153-81&rft.date=1982&rft.aulast=Harris&rft.aufirst=John&rft_id=http%3A%2F%2Fwww1.aiatsis.gov.au%2Fexhibitions%2Fe_access%2Fserial%2Fm0029743_v_a.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text">Dawson, J. "<a rel="nofollow" class="external text" href="https://archive.org/details/australianabori00dawsgoog"><i>Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria</i></a> (1881), p. xcviii.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatsushita1998" class="citation conference cs1">Matsushita, Shuji (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html"><i>Decimal vs. Duodecimal: An interaction between two systems of numeration</i></a>. 2nd Meeting of the AFLANG, October 1998, Tokyo. Archived from <a rel="nofollow" class="external text" href="http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html">the original</a> on 2008-10-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-05-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Decimal+vs.+Duodecimal%3A+An+interaction+between+two+systems+of+numeration&rft.date=1998&rft.aulast=Matsushita&rft.aufirst=Shuji&rft_id=http%3A%2F%2Fwww3.aa.tufs.ac.jp%2F~P_aflang%2FTEXTS%2Foct98%2Fdecimal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMazaudon2002" class="citation book cs1">Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques (ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160328145817/http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf"><i>La Pluralité</i></a> <span class="cs1-format">(PDF)</span>. Leuven: Peeters. pp. 91–119. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-429-1295-2" title="Special:BookSources/90-429-1295-2"><bdi>90-429-1295-2</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-28<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-09-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Les+principes+de+construction+du+nombre+dans+les+langues+tib%C3%A9to-birmanes&rft.btitle=La+Pluralit%C3%A9&rft.place=Leuven&rft.pages=91-119&rft.pub=Peeters&rft.date=2002&rft.isbn=90-429-1295-2&rft.aulast=Mazaudon&rft.aufirst=Martine&rft_id=http%3A%2F%2Flacito.vjf.cnrs.fr%2Fdocuments%2Fpubli%2Fnum_WEB.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheetham1978" class="citation journal cs1">Cheetham, Brian (1978). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070928061238/http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm">"Counting and Number in Huli"</a>. <i>Papua New Guinea Journal of Education</i>. <b>14</b>: 16–35. Archived from <a rel="nofollow" class="external text" href="http://www.uog.ac.pg/PUB08-Oct-03/cheetham.htm">the original</a> on 2007-09-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Papua+New+Guinea+Journal+of+Education&rft.atitle=Counting+and+Number+in+Huli&rft.volume=14&rft.pages=16-35&rft.date=1978&rft.aulast=Cheetham&rft.aufirst=Brian&rft_id=http%3A%2F%2Fwww.uog.ac.pg%2FPUB08-Oct-03%2Fcheetham.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBowersLepi1975" class="citation journal cs1">Bowers, Nancy; Lepi, Pundia (1975). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110604091351/http://www.ethnomath.org/resources/bowers-lepi1975.pdf">"Kaugel Valley systems of reckoning"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the Polynesian Society</i>. <b>84</b> (3): 309–24. Archived from <a rel="nofollow" class="external text" href="http://www.ethnomath.org/resources/bowers-lepi1975.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-06-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Polynesian+Society&rft.atitle=Kaugel+Valley+systems+of+reckoning&rft.volume=84&rft.issue=3&rft.pages=309-24&rft.date=1975&rft.aulast=Bowers&rft.aufirst=Nancy&rft.au=Lepi%2C+Pundia&rft_id=http%3A%2F%2Fwww.ethnomath.org%2Fresources%2Fbowers-lepi1975.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOwens2001" class="citation cs2">Owens, Kay (2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150926003303/http://www.uog.ac.pg/glec/Key/Kay/owens131.htm">"The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania"</a>, <i>Mathematics Education Research Journal</i>, <b>13</b> (1): 47–71, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001MEdRJ..13...47O">2001MEdRJ..13...47O</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03217098">10.1007/BF03217098</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:161535519">161535519</a>, archived from <a rel="nofollow" class="external text" href="http://www.uog.ac.pg/glec/Key/Kay/owens131.htm">the original</a> on 2015-09-26</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Education+Research+Journal&rft.atitle=The+Work+of+Glendon+Lean+on+the+Counting+Systems+of+Papua+New+Guinea+and+Oceania&rft.volume=13&rft.issue=1&rft.pages=47-71&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A161535519%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF03217098&rft_id=info%3Abibcode%2F2001MEdRJ..13...47O&rft.aulast=Owens&rft.aufirst=Kay&rft_id=http%3A%2F%2Fwww.uog.ac.pg%2Fglec%2FKey%2FKay%2Fowens131.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecimal" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Orders_of_magnitude" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Orders_of_magnitude" title="Template:Orders of magnitude"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Orders_of_magnitude" title="Template talk:Orders of magnitude"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Orders_of_magnitude" title="Special:EditPage/Template:Orders of 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href="/wiki/Computer_performance_by_orders_of_magnitude" title="Computer performance by orders of magnitude">Computing</a></li> <li><a href="/wiki/Orders_of_magnitude_(current)" title="Orders of magnitude (current)">Current</a></li> <li><a href="/wiki/Orders_of_magnitude_(data)" title="Orders of magnitude (data)">Data</a></li> <li><a href="/wiki/Orders_of_magnitude_(density)" class="mw-redirect" title="Orders of magnitude (density)">Density</a></li> <li><a href="/wiki/Orders_of_magnitude_(energy)" title="Orders of magnitude (energy)">Energy</a> / <a href="/wiki/Energy_density_Extended_Reference_Table" title="Energy density Extended Reference Table">Energy density</a></li> <li><a href="/wiki/Orders_of_magnitude_(entropy)" title="Orders of magnitude (entropy)">Entropy</a></li> <li><a href="/wiki/Orders_of_magnitude_(force)" title="Orders of magnitude (force)">Force</a></li> <li><a href="/wiki/Orders_of_magnitude_(frequency)" title="Orders of magnitude (frequency)">Frequency</a></li> <li><a href="/wiki/Orders_of_magnitude_(illuminance)" title="Orders of magnitude (illuminance)">Illuminance</a></li> <li><a href="/wiki/Orders_of_magnitude_(length)" title="Orders of magnitude (length)">Length</a></li> <li><a href="/wiki/Orders_of_magnitude_(luminance)" class="mw-redirect" title="Orders of magnitude (luminance)">Luminance</a></li> <li><a href="/wiki/Orders_of_magnitude_(magnetic_moment)" title="Orders of magnitude (magnetic moment)">Magnetic moment</a></li> <li><a href="/wiki/Orders_of_magnitude_(magnetic_field)" title="Orders of magnitude (magnetic field)">Magnetic field</a></li> <li><a href="/wiki/Orders_of_magnitude_(mass)" title="Orders of magnitude (mass)">Mass</a></li> <li><a href="/wiki/Orders_of_magnitude_(molar_concentration)" title="Orders of magnitude (molar concentration)">Molarity</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)" title="Orders of magnitude (numbers)">Numbers</a></li> <li><a href="/wiki/Orders_of_magnitude_(power)" title="Orders of magnitude (power)">Power</a></li> <li><a href="/wiki/Orders_of_magnitude_(pressure)" title="Orders of magnitude (pressure)">Pressure</a></li> <li><a href="/wiki/Orders_of_magnitude_(probability)" title="Orders of magnitude (probability)">Probability</a></li> <li><a href="/wiki/Orders_of_magnitude_(radiation)" title="Orders of magnitude (radiation)">Radiation</a></li> <li><a href="/wiki/Sound_pressure#Examples_of_sound_pressure" title="Sound pressure">Sound pressure</a></li> <li><a href="/wiki/Orders_of_magnitude_(specific_heat_capacity)" title="Orders of magnitude (specific heat capacity)">Specific heat capacity</a></li> <li><a href="/wiki/Orders_of_magnitude_(speed)" title="Orders of magnitude (speed)">Speed</a></li> <li><a href="/wiki/Orders_of_magnitude_(temperature)" title="Orders of magnitude (temperature)">Temperature</a></li> <li><a href="/wiki/Orders_of_magnitude_(torque)" title="Orders of magnitude (torque)">Torque</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Time</a></li> <li><a href="/wiki/Orders_of_magnitude_(voltage)" title="Orders of magnitude (voltage)">Voltage</a></li> <li><a href="/wiki/Orders_of_magnitude_(volume)" title="Orders of magnitude (volume)">Volume</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Back-of-the-envelope_calculation" title="Back-of-the-envelope calculation">Back-of-the-envelope calculation</a></li> <li><a href="/wiki/Electronics_industry#List_of_best-selling_electronic_devices" title="Electronics industry">Best-selling electronic devices</a></li> <li><a href="/wiki/Fermi_problem" title="Fermi problem">Fermi problem</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a> and <a href="/wiki/Decade_(log_scale)" title="Decade (log scale)">decades</a> <ul><li><a class="mw-selflink selflink">10th</a></li> <li><a href="/wiki/Hundredth" title="Hundredth">100th</a></li> <li><a href="/wiki/Millionth" title="Millionth">1000000th</a></li></ul></li> <li><a href="/wiki/Metric_prefix" title="Metric prefix">Metric (SI) prefix</a></li> <li><a href="/wiki/Macroscopic_scale" title="Macroscopic scale">Macroscopic scale</a></li> <li><a href="/wiki/Microscopic_scale" title="Microscopic scale">Microscopic scale</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_system_of_units" title="Astronomical system of units">Astronomical system of units</a></li> <li><a href="/wiki/Location_of_Earth" title="Location of Earth">Earth's location in the Universe</a></li> <li><i><a href="/wiki/Cosmic_View" title="Cosmic View">Cosmic View</a></i> (1957 book)</li> <li><i><a href="/wiki/To_the_Moon_and_Beyond" title="To the Moon and Beyond">To the Moon and Beyond</a></i> (1964 film)</li> <li><i><a href="/wiki/Cosmic_Zoom" title="Cosmic Zoom">Cosmic Zoom</a></i> (1968 film)</li> <li><a href="/wiki/Powers_of_Ten_(film)" title="Powers of Ten (film)"><i>Powers of Ten</i> (1968 and 1977 films)</a></li> <li><a href="/wiki/Cosmic_Voyage_(1996_film)" title="Cosmic Voyage (1996 film)"><i>Cosmic Voyage</i> (1996 documentary)</a></li> <li><i><a href="/wiki/The_Scale_of_the_Universe" title="The Scale of the Universe">The Scale of the Universe</a></i> (2010)</li> <li><i><a href="/wiki/Cosmic_Eye" title="Cosmic Eye">Cosmic Eye</a></i> (2012)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" 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