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<li id="toc-Other_historical_numeral_systems_using_digits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_historical_numeral_systems_using_digits"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Other historical numeral systems using digits</span> </div> </a> <ul id="toc-Other_historical_numeral_systems_using_digits-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_numeral_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Main_numeral_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Main numeral systems</span> </div> </a> <ul id="toc-Main_numeral_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Positional_systems_in_detail" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Positional_systems_in_detail"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Positional systems in detail</span> </div> </a> <ul id="toc-Positional_systems_in_detail-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_variable-length_integers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalized_variable-length_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generalized variable-length integers</span> </div> </a> <ul id="toc-Generalized_variable-length_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Numeral system</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 82 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-82" 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">82 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/Zahlensystem" title="Zahlensystem – Alemannic" lang="gsw" hreflang="gsw" data-title="Zahlensystem" 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" title="نظام عد – Arabic" lang="ar" hreflang="ar" data-title="نظام عد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Say_sisteml%C9%99ri" title="Say sistemləri – Azerbaijani" lang="az" hreflang="az" data-title="Say sistemləri" 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%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AA%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BF" title="সংখ্যাপদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="সংখ্যাপদ্ধতি" 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/S%C3%B2%CD%98-j%C4%AB" title="Sò͘-jī – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-jī" 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-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%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="Сістэма злічэння – Belarusian" lang="be" hreflang="be" data-title="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%A1%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="Сыстэма зьлічэньня – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%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="Бройна система – Bulgarian" lang="bg" hreflang="bg" data-title="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/Brojevni_sistem" title="Brojevni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Brojevni sistem" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_de_numeraci%C3%B3" title="Sistema de numeració – Catalan" lang="ca" hreflang="ca" data-title="Sistema de numeració" 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%A8%D1%83%D1%82_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Шут тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="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/%C4%8C%C3%ADseln%C3%A1_soustava" title="Číselná soustava – Czech" lang="cs" hreflang="cs" data-title="Číselná 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/Tsika_yekurava_nhamba" title="Tsika yekurava nhamba – Shona" lang="sn" hreflang="sn" data-title="Tsika yekurava nhamba" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Talsystem" title="Talsystem – Danish" lang="da" hreflang="da" data-title="Talsystem" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zahlensystem" title="Zahlensystem – German" lang="de" hreflang="de" data-title="Zahlensystem" 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/Arvus%C3%BCsteem" title="Arvusüsteem – Estonian" lang="et" hreflang="et" data-title="Arvusüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_de_numeraci%C3%B3n" title="Sistema de numeración – Spanish" lang="es" hreflang="es" data-title="Sistema de numeración" 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/Nombrosistemo" title="Nombrosistemo – Esperanto" lang="eo" hreflang="eo" data-title="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" title="Zenbaki-sistema – Basque" lang="eu" hreflang="eu" data-title="Zenbaki-sistema" 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%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D8%B4%D9%85%D8%A7%D8%B1%D8%B4" title="دستگاه شمارش – Persian" lang="fa" hreflang="fa" data-title="دستگاه شمارش" 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_de_num%C3%A9ration" title="Système de numération – French" lang="fr" hreflang="fr" data-title="Système de numération" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_numeraci%C3%B3n" title="Sistema de numeración – Galician" lang="gl" hreflang="gl" data-title="Sistema de numeración" 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/%EA%B8%B0%EC%88%98%EB%B2%95" title="기수법 – Korean" lang="ko" hreflang="ko" data-title="기수법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B7%D5%BE%D5%A1%D6%80%D5%AF%D5%B4%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Հաշվարկման համակարգ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Հաշվարկման համակարգ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF%E0%A4%AF%E0%A4%BE%E0%A4%81" title="संख्या पद्धतियाँ – Hindi" lang="hi" hreflang="hi" data-title="संख्या पद्धतियाँ" 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/Brojevni_sustav" title="Brojevni sustav – Croatian" lang="hr" hreflang="hr" data-title="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/Nombrosistemo" title="Nombrosistemo – Ido" lang="io" hreflang="io" data-title="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" title="Sistem bilangan – Indonesian" lang="id" hreflang="id" data-title="Sistem bilangan" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Talnakerfi" title="Talnakerfi – Icelandic" lang="is" hreflang="is" data-title="Talnakerfi" 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_di_numerazione" title="Sistema di numerazione – Italian" lang="it" hreflang="it" data-title="Sistema di numerazione" 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%A9%D7%99%D7%98%D7%AA_%D7%A1%D7%A4%D7%99%D7%A8%D7%94" title="שיטת ספירה – Hebrew" lang="he" hreflang="he" data-title="שיטת ספירה" 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" title="Sistem wilangan – Javanese" lang="jv" hreflang="jv" data-title="Sistem wilangan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%95%E1%83%9A%E1%83%98%E1%83%A1_%E1%83%A1%E1%83%98%E1%83%A1%E1%83%A2%E1%83%94%E1%83%9B%E1%83%90" title="თვლის სისტემა – Georgian" lang="ka" hreflang="ka" data-title="თვლის სისტემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BD%D0%B0%D1%83_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Санау жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="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_nimewotasyon" title="Sistèm nimewotasyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm nimewotasyon" 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%AD%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="Эсептөө тутуму – Kyrgyz" lang="ky" hreflang="ky" data-title="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_numerale" title="Systema numerale – Latin" lang="la" hreflang="la" data-title="Systema numerale" 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/Skait%C4%AB%C5%A1anas_sist%C4%93ma" title="Skaitīšanas sistēma – Latvian" lang="lv" hreflang="lv" data-title="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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%A1mrendszer" title="Számrendszer – Hungarian" lang="hu" hreflang="hu" data-title="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%91%D1%80%D0%BE%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Броен систем – Macedonian" lang="mk" hreflang="mk" data-title="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%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE%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%99%E0%B5%8D%E0%B4%99%E0%B5%BE" title="സംഖ്യാസമ്പ്രദായങ്ങൾ – Malayalam" lang="ml" hreflang="ml" data-title="സംഖ്യാസമ്പ്രദായങ്ങൾ" 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/Sistem_angka" title="Sistem angka – Malay" lang="ms" hreflang="ms" data-title="Sistem angka" 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-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%83%EA%AF%81%EA%AF%A4%EA%AF%A1" title="ꯃꯁꯤꯡ – Manipuri" lang="mni" hreflang="mni" data-title="ꯃꯁꯤꯡ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="Manipuri" class="interlanguage-link-target"><span>ꯃꯤꯇꯩ ꯂꯣꯟ</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sistema_de_numera%C3%A7on" title="Sistema de numeraçon – Mirandese" lang="mwl" hreflang="mwl" data-title="Sistema de numeraçon" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%8F%E1%80%94%E1%80%BA%E1%80%B8%E1%80%81%E1%80%BC%E1%80%B1" title="ဂဏန်းခြေ – Burmese" lang="my" hreflang="my" data-title="ဂဏန်းခြေ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Talstelsel" title="Talstelsel – Dutch" lang="nl" hreflang="nl" data-title="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%91%BD%E6%95%B0%E6%B3%95" title="命数法 – Japanese" lang="ja" hreflang="ja" data-title="命数法" 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/Tallsystem" title="Tallsystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tallsystem" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_de_numeracion" title="Sistèma de numeracion – Occitan" lang="oc" hreflang="oc" data-title="Sistèma de numeracion" 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%A7%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="Чотрадам системе – Eastern Mari" lang="mhr" hreflang="mhr" data-title="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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sanoq_sistemasi" title="Sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Sanoq sistemasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%A3%E0%A8%BE%E0%A8%B2%E0%A9%80" title="ਸੰਖਿਆ ਪ੍ਰਣਾਲੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਸੰਖਿਆ ਪ੍ਰਣਾਲੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF%DB%8C_%D9%86%D8%B8%D8%A7%D9%85" title="عددی نظام – Western Punjabi" lang="pnb" hreflang="pnb" data-title="عددی نظام" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/System_liczbowy" title="System liczbowy – Polish" lang="pl" hreflang="pl" data-title="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" title="Sistema de numeração – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de numeração" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Sanaq_sistemas%C4%B1" title="Sanaq sisteması – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Sanaq sisteması" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_de_numera%C8%9Bie" title="Sistem de numerație – Romanian" lang="ro" hreflang="ro" data-title="Sistem de numerație" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%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="Система счисления – Russian" lang="ru" hreflang="ru" data-title="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/Lebadi" title="Lebadi – Northern Sotho" lang="nso" hreflang="nso" data-title="Lebadi" 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-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%AD_%E0%B6%B4%E0%B6%AF%E0%B7%8A%E0%B6%B0%E0%B6%AD%E0%B7%92" title="සංඛ්‍යාත පද්ධති – Sinhala" lang="si" hreflang="si" data-title="සංඛ්‍යාත පද්ධති" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Numeral_system" title="Numeral system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Numeral system" 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/%C4%8C%C3%ADseln%C3%A1_s%C3%BAstava" title="Číselná sústava – Slovak" lang="sk" hreflang="sk" data-title="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/%C5%A0tevilski_sistem" title="Številski sistem – Slovenian" lang="sl" hreflang="sl" data-title="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%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%DA%98%D9%85%D8%A7%D8%B1%D8%A7%DA%B5" title="سیستمی ژماراڵ – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سیستمی ژماراڵ" 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%91%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Бројевни систем – Serbian" lang="sr" hreflang="sr" data-title="Бројевни 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/Brojevni_sistem" title="Brojevni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Brojevni 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/Lukuj%C3%A4rjestelm%C3%A4" title="Lukujärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Lukujärjestelmä" 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/Talsystem" title="Talsystem – Swedish" lang="sv" hreflang="sv" data-title="Talsystem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Numerasyon" title="Numerasyon – Tagalog" lang="tl" hreflang="tl" data-title="Numerasyon" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%B1%E0%AE%BF_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="எண்குறி முறைமை – Tamil" lang="ta" hreflang="ta" data-title="எண்குறி முறைமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%86%E0%B0%B2%E0%B1%81%E0%B0%97%E0%B1%81" title="తెలుగు – Telugu" lang="te" hreflang="te" data-title="తెలుగు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B9%80%E0%B8%A5%E0%B8%82" title="ระบบเลข – Thai" lang="th" hreflang="th" data-title="ระบบเลข" 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/Say%C4%B1sal_sistem" title="Sayısal sistem – Turkish" lang="tr" hreflang="tr" data-title="Sayısal sistem" 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%A1%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="Система числення – Ukrainian" lang="uk" hreflang="uk" data-title="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/%D8%B9%D8%AF%D8%AF%DB%8C_%D9%86%D8%B8%D8%A7%D9%85" title="عددی نظام – Urdu" lang="ur" hreflang="ur" data-title="عددی نظام" 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_%C4%91%E1%BA%BFm" title="Hệ đếm – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ đếm" 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/%E8%A1%A8%E6%95%B8%E6%B3%95" title="表數法 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="表數法" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_pag-ihap" title="Sistema pag-ihap – Waray" lang="war" hreflang="war" data-title="Sistema pag-ihap" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F" title="记数系统 – Wu" lang="wuu" hreflang="wuu" data-title="记数系统" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%95%D7%9E%D7%A2%D7%A8%D7%9F_%D7%A1%D7%99%D7%A1%D7%98%D7%A2%D7%9D" title="נומערן סיסטעם – Yiddish" lang="yi" hreflang="yi" data-title="נומערן סיסטעם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A8%98%E6%95%B8%E6%B3%95" title="記數法 – Cantonese" lang="yue" hreflang="yue" data-title="記數法" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F" title="记数系统 – Chinese" lang="zh" hreflang="zh" data-title="记数系统" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q122653#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Numeral_system" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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.sidebar:not(.notheme) .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 class="mw-selflink selflink">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 href="/wiki/Decimal" title="Decimal">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> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Numeral_Systems_of_the_World.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Numeral_Systems_of_the_World.svg/264px-Numeral_Systems_of_the_World.svg.png" decoding="async" width="264" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Numeral_Systems_of_the_World.svg/396px-Numeral_Systems_of_the_World.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Numeral_Systems_of_the_World.svg/528px-Numeral_Systems_of_the_World.svg.png 2x" data-file-width="512" data-file-height="312" /></a><figcaption>Numbers written in different numeral systems</figcaption></figure> <p>A <b>numeral system</b> is a writing system for expressing numbers; that is, a <a href="/wiki/Mathematical_notation" title="Mathematical notation">mathematical notation</a> for representing numbers of a given set, using <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> or other symbols in a consistent manner. </p><p>The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number <i>eleven</i> in the <a href="/wiki/Decimal" title="Decimal">decimal or base-10</a> numeral system (today, the most common system globally), the number <i>three</i> in the <a href="/wiki/Binary_number" title="Binary number">binary or base-2</a> numeral system (used in modern computers), and the number <i>two</i> in the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a> (used in <a href="/wiki/Tally_marks" title="Tally marks">tallying</a> scores). </p><p>The number the numeral represents is called its value. Not all number systems can represent the same set of numbers; for example, <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a> cannot represent the number zero. </p><p>Ideally, a numeral system will: </p> <ul><li>Represent a useful set of numbers (e.g. all <a href="/wiki/Integer" title="Integer">integers</a>, or <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>)</li> <li>Give every number represented a unique representation (or at least a standard representation)</li> <li>Reflect the <a href="/wiki/Algebra" title="Algebra">algebraic</a> and <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> structure of the numbers.</li></ul> <p>For example, the usual <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a> gives every nonzero <a href="/wiki/Natural_number" title="Natural number">natural number</a> a unique representation as a finite <a href="/wiki/Sequence" title="Sequence">sequence</a> of digits, beginning with a non-zero digit. </p><p>Numeral systems are sometimes called <i><a href="/wiki/Number_system" class="mw-redirect" title="Number system">number systems</a></i>, but that name is ambiguous, as it could refer to different systems of numbers, such as the system of <a href="/wiki/Real_number" title="Real number">real numbers</a>, the system of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, various <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex number</a> systems, the system of <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>, etc. Such systems are, however, not the topic of this article. </p> <meta property="mw:PageProp/toc" /> <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=Numeral_system&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Numerical_digit#History" title="Numerical digit">Numerical digit § History</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Numeral_system&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">July 2024</span>)</i></span></div></td></tr></tbody></table> <div style="float:right;"> <table class="wikitable zebra"> <tbody><tr> <th>Western Arabic </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9 </td></tr> <tr> <th>Eastern Arabic </th> <td>٠</td> <td>١</td> <td>٢</td> <td>٣</td> <td>٤</td> <td>٥</td> <td>٦</td> <td>٧</td> <td>٨</td> <td>٩ </td></tr> <tr> <th>Persian </th> <td>۰</td> <td>۱</td> <td>۲</td> <td>۳</td> <td>۴</td> <td>۵</td> <td>۶</td> <td>۷</td> <td>۸</td> <td>۹ </td></tr> <tr> <th>Devanagari </th> <td>०</td> <td>१</td> <td>२</td> <td>३</td> <td>४</td> <td>५</td> <td>६</td> <td>७</td> <td>८</td> <td>९ </td></tr></tbody></table> </div> <p>The first true written <a href="/wiki/Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral system</a> is considered to be the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>. This system was established by the 7th century in India,<sup id="cite_ref-O'Connor_and_Robertson_1-0" class="reference"><a href="#cite_note-O'Connor_and_Robertson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> but was not yet in its modern form because the use of the digit <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The original numerals were very similar to the modern ones, even down to the <a href="/wiki/Glyph" title="Glyph">glyphs</a> used to represent digits.<sup id="cite_ref-O'Connor_and_Robertson_1-1" class="reference"><a href="#cite_note-O'Connor_and_Robertson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Maya.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/150px-Maya.svg.png" decoding="async" width="150" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/225px-Maya.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/300px-Maya.svg.png 2x" data-file-width="250" data-file-height="320" /></a><figcaption>The digits of the Maya numeral system</figcaption></figure> <p>By the 13th century, <a href="/wiki/Western_Arabic_numerals" class="mw-redirect" title="Western Arabic numerals">Western Arabic numerals</a> were accepted in European mathematical circles (<a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> used them in his <span title="Latin-language text"><i lang="la"><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></span>). They began to enter common use in the 15th century.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures. </p> <div class="mw-heading mw-heading3"><h3 id="Other_historical_numeral_systems_using_digits">Other historical numeral systems using digits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=2" title="Edit section: Other historical numeral systems using digits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The exact age of the <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a> is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was <a href="/wiki/Vigesimal" title="Vigesimal">vigesimal</a> (base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The <a href="/wiki/Mayas" class="mw-redirect" title="Mayas">Mayas</a> had no equivalent of the modern <a href="/wiki/Decimal_separator" title="Decimal separator">decimal separator</a>, so their system could not represent fractions.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2024)">citation needed</span></a></i>]</sup> </p><p>The <a href="/wiki/Thai_numerals" title="Thai numerals">Thai numeral system</a> is identical to the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> except for the symbols used to represent digits. The use of these digits is less common in <a href="/wiki/Thailand" title="Thailand">Thailand</a> than it once was, but they are still used alongside Arabic numerals.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2024)">citation needed</span></a></i>]</sup> </p><p>The rod numerals, the written forms of <a href="/wiki/Counting_rods" title="Counting rods">counting rods</a> once used by <a href="/wiki/China" title="China">Chinese</a> and <a href="/wiki/Japan" title="Japan">Japanese</a> mathematicians, are a decimal positional system used for performing decimal calculations. Rods were placed on a counting board and slid forwards or backwards to change the decimal place. The <i><a href="/wiki/Sunzi_Suanjing" title="Sunzi Suanjing">Sūnzĭ Suànjīng</a></i>, a mathematical treatise dated to between the 3rd and 5th centuries AD, provides detailed instructions for the system, which is thought to have been in use since at least the 4th century BC.<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> Zero was not initially treated as a number, but as a vacant position.<sup id="cite_ref-Crossley_5-0" class="reference"><a href="#cite_note-Crossley-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Later sources introduced conventions for the expression of zero and negative numbers. The use of a round symbol <span title="Chinese-language text"><span lang="zh">〇</span></span> for zero is first attested in the <i><a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a></i> of 1247 AD.<sup id="cite_ref-Qin_6-0" class="reference"><a href="#cite_note-Qin-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The origin of this symbol is unknown; it may have been produced by modifying a square symbol.<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> The <a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou numerals</a>, a descendant of rod numerals, are still used today for some commercial purposes.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2024)">citation needed</span></a></i>]</sup> </p> <table class="wikitable" style="text-align:center"> <caption>Rod numerals (vertical) </caption> <tbody><tr> <th style="width:50px">0 </th> <th style="width:50px">1 </th> <th style="width:50px">2 </th> <th style="width:50px">3 </th> <th style="width:50px">4 </th> <th style="width:50px">5 </th> <th style="width:50px">6 </th> <th style="width:50px">7 </th> <th style="width:50px">8 </th> <th style="width:50px">9 </th></tr> <tr> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b5/Counting_rod_v3.png" decoding="async" width="19" height="29" class="mw-file-element" data-file-width="19" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_v5.png" decoding="async" width="31" height="29" class="mw-file-element" data-file-width="31" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/Counting_rod_v9.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td></tr> <tr> <th style="width:50px">–0 </th> <th style="width:50px">–1 </th> <th style="width:50px">–2 </th> <th style="width:50px">–3 </th> <th style="width:50px">–4 </th> <th style="width:50px">–5 </th> <th style="width:50px">–6 </th> <th style="width:50px">–7 </th> <th style="width:50px">–8 </th> <th style="width:50px">–9 </th></tr> <tr> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_-0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/37/Counting_rod_-0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3f/Counting_rod_v-3.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/ba/Counting_rod_v-4.png" decoding="async" width="31" height="29" class="mw-file-element" data-file-width="31" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/66/Counting_rod_v-5.png" decoding="async" width="35" height="29" class="mw-file-element" data-file-width="35" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/11/Counting_rod_v-6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/04/Counting_rod_v-7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Counting_rod_v-8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0d/Counting_rod_v-9.png" decoding="async" width="31" height="32" class="mw-file-element" data-file-width="31" data-file-height="32" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Main_numeral_systems">Main numeral systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=3" title="Edit section: Main numeral systems"><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/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></div> <p>The most commonly used system of numerals is <a href="/wiki/Decimal" title="Decimal">decimal</a>. <a href="/wiki/Indian_mathematicians" class="mw-redirect" title="Indian mathematicians">Indian mathematicians</a> are credited with developing the integer version, the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> of <a href="/wiki/Patna" title="Patna">Kusumapura</a> developed the <a href="/wiki/Place-value_notation" class="mw-redirect" title="Place-value notation">place-value notation</a> in the 5th century and a century later <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> introduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematician <a href="/wiki/Abu%27l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a> in 952–953, and the decimal point notation was introduced<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (February 2021)">when?</span></a></i>]</sup> by <a href="/wiki/Sind_ibn_Ali" class="mw-redirect" title="Sind ibn Ali">Sind ibn Ali</a>, who also wrote the earliest treatise on Arabic numerals. The Hindu–Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, as they learned them from the Arabs. </p><p>The simplest numeral system is the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a>, in which every <a href="/wiki/Natural_number" title="Natural number">natural number</a> is represented by a corresponding number of symbols. If the symbol <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">/</span> is chosen, for example, then the number seven would be represented by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">///////</span>. <a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a> represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>. <a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Elias gamma coding</a>, which is commonly used in <a href="/wiki/Data_compression" title="Data compression">data compression</a>, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral. </p><p>The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">+++ ////</span> and the number 123 as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">+ − − ///</span> without any need for zero. This is called <a href="/wiki/Sign-value_notation" title="Sign-value notation">sign-value notation</a>. The ancient <a href="/wiki/Egyptian_numeral_system" class="mw-redirect" title="Egyptian numeral system">Egyptian numeral system</a> was of this type, and the <a href="/wiki/Roman_numeral_system" class="mw-redirect" title="Roman numeral system">Roman numeral system</a> was a modification of this idea. </p><p>More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, one could then write C+ D/ for the number 304 (the number of these abbreviations is sometimes called the <i>base</i> of the system). This system is used when writing <a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese numerals</a> and other East Asian numerals based on Chinese. The number system of the English language is of this type ("three hundred [and] four"), as are those of other spoken languages, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is <i>soixante dix-neuf</i> (<span class="nowrap">60 + 10 + 9</span>) and in Welsh is <i>pedwar ar bymtheg a thrigain</i> (<span class="nowrap">4 + (5 + 10) + (3 × 20)</span>) or (somewhat archaic) <i>pedwar ugain namyn un</i> (<span class="nowrap">4 × 20 − 1</span>). In English, one could say "four score less one", as in the famous <a href="/wiki/Gettysburg_Address" title="Gettysburg Address">Gettysburg Address</a> representing "87 years ago" as "four score and seven years ago". </p><p>More elegant is a <i><a href="/wiki/Positional_notation" title="Positional notation">positional system</a></i>, also known as place-value notation. The positional systems are classified by their <i>base</i> or <i><a href="/wiki/Radix" title="Radix">radix</a></i>, which is the number of symbols called <i><a href="/wiki/Numerical_digit" title="Numerical digit">digits</a></i> used by the system. In base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in <span class="nowrap">304 = 3×100 + 0×10 + 4×1</span> or more precisely <span class="nowrap">3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup></span>. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system. </p><p>Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The positional decimal system is presently universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers. </p><p>In computers, the main numeral systems are based on the positional system in base 2 (<a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a>), with two <a href="/wiki/Binary_digit" class="mw-redirect" title="Binary digit">binary digits</a>, 0 and 1. Positional systems obtained by grouping binary digits by three (<a href="/wiki/Octal_numeral_system" class="mw-redirect" title="Octal numeral system">octal numeral system</a>) or four (<a href="/wiki/Hexadecimal_numeral_system" class="mw-redirect" title="Hexadecimal numeral system">hexadecimal numeral system</a>) are commonly used. For very large integers, bases 2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the <a href="/wiki/Machine_word" class="mw-redirect" title="Machine word">machine word</a>) are used, as, for example, in <a href="/wiki/GNU_Multiple_Precision_Arithmetic_Library" title="GNU Multiple Precision Arithmetic Library">GMP</a>. </p><p>In certain biological systems, the <a href="/wiki/Unary_coding" title="Unary coding">unary coding</a> system is employed. Unary numerals used in the <a href="/wiki/Neural_circuit" title="Neural circuit">neural circuits</a> responsible for <a href="/wiki/Birdsong" class="mw-redirect" title="Birdsong">birdsong</a> production.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (<a href="/wiki/High_vocal_center" class="mw-redirect" title="High vocal center">high vocal center</a>). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness. </p><p>The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the <a href="/wiki/Arithmetic_sequence" class="mw-redirect" title="Arithmetic sequence">arithmetic</a> numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the <a href="/wiki/Geometric_sequence" class="mw-redirect" title="Geometric sequence">geometric</a> numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the <a href="/wiki/Greek_numerals" title="Greek numerals">Ionic system</a>), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses <i>both</i> arithmetic and geometric numerals. </p><p>In some areas of computer science, a modified base <i>k</i> positional system is used, called <a href="/wiki/Bijective_numeration" title="Bijective numeration">bijective numeration</a>, with digits 1, 2, ..., <i>k</i> (<span class="nowrap"><i>k</i> ≥ 1</span>), and zero being represented by an empty string. This establishes a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-<i>k</i> numeration is also called <i>k</i>-adic notation, not to be confused with <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>. Bijective base 1 is the same as unary. </p> <div class="mw-heading mw-heading2"><h2 id="Positional_systems_in_detail">Positional systems in detail</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=4" title="Edit section: Positional systems in detail"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a></div> <p>In a positional base <i>b</i> numeral system (with <i>b</i> a <a href="/wiki/Natural_number" title="Natural number">natural number</a> greater than 1 known as the <a href="/wiki/Radix" title="Radix">radix</a> or <i>base</i> of the system), <i>b</i> basic symbols (or digits) corresponding to the first <i>b</i> natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by <i>b</i>. </p><p>For example, in the <a href="/wiki/Decimal" title="Decimal">decimal</a> system (base 10), the numeral 4327 means <span class="texhtml">(<b>4</b>×10<sup>3</sup>) + (<b>3</b>×10<sup>2</sup>) + (<b>2</b>×10<sup>1</sup>) + (<b>7</b>×10<sup>0</sup>)</span>, noting that <span class="texhtml">10<sup>0</sup> = 1</span>. </p><p>In general, if <i>b</i> is the base, one writes a number in the numeral system of base <i>b</i> by expressing it in the form <span class="texhtml"><i>a</i><sub><i>n</i></sub><i>b</i><sup><i>n</i></sup> + <i>a</i><sub><i>n</i> − 1</sub><i>b</i><sup><i>n</i> − 1</sup> + <i>a</i><sub><i>n</i> − 2</sub><i>b</i><sup><i>n</i> − 2</sup> + ... + <i>a</i><sub>0</sub><i>b</i><sup>0</sup></span> and writing the enumerated digits <span class="texhtml"><i>a</i><sub><i>n</i></sub><i>a</i><sub><i>n</i> − 1</sub><i>a</i><sub><i>n</i> − 2</sub> ... <i>a</i><sub>0</sub></span> in descending order. The digits are natural numbers between 0 and <span class="texhtml"><i>b</i> − 1</span>, inclusive. </p><p>If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number<sub>base</sub>. Unless specified by context, numbers without subscript are considered to be decimal. </p><p>By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes <span class="texhtml">1×2<sup>1</sup> + 0×2<sup>0</sup> + 1×2<sup>−1</sup> + 1×2<sup>−2</sup> = 2.75</span>. </p><p>In general, numbers in the base <i>b</i> system are of the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed875ba981decb322a05335f7efdb5490244d67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.072ex; height:7.009ex;" alt="{\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}.}"></span></dd></dl> <p>The numbers <i>b</i><sup><i>k</i></sup> and <i>b</i><sup>−<i>k</i></sup> are the <a href="/wiki/Weight_function" title="Weight function">weights</a> of the corresponding digits. The position <i>k</i> is the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the corresponding weight <i>w</i>, that is <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 k=\log _{b}w=\log _{b}b^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>w</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\log _{b}w=\log _{b}b^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fe6f9d7c4f8a8275a0c41afb2bfd406150d4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.752ex; height:3.176ex;" alt="{\displaystyle k=\log _{b}w=\log _{b}b^{k}}"></span>. The highest used position is close to the <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order of magnitude</a> of the number. </p><p>The number of <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a> required in the <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary numeral system</a> for <i>describing the weight</i> would have been <b>w</b>. In the positional system, the number of digits required to describe it is only <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 k+1=\log _{b}w+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>w</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k+1=\log _{b}w+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa94ae36727c9da610fe1b54cc4f5ea559d77aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.276ex; height:2.676ex;" alt="{\displaystyle k+1=\log _{b}w+1}"></span>, for <i>k</i> ≥ 0. For example, to describe the weight 1000 then four digits are needed because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}1000+1=3+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>1000</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}1000+1=3+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276fbacb3b57e34c84dd39e5c725d1ce6e554f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.152ex; height:2.676ex;" alt="{\displaystyle \log _{10}1000+1=3+1}"></span>. The number of digits required to <i>describe the position</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>w</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1772ff8f82c45f3dc17e901a92d6c80d5d80985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.869ex; height:2.676ex;" alt="{\displaystyle \log _{b}k+1=\log _{b}\log _{b}w+1}"></span> (in positions 1, 10, 100,... only for simplicity in the decimal example). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left right right right right right right right" rowspacing="4pt" columnspacing="1em" rowlines="solid none solid none" columnlines="solid none none none none none none"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Position</mtext> </mrow> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Weight</mtext> </mrow> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Digit</mtext> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Decimal example weight</mtext> </mrow> </mtd> <mtd> <mn>1000</mn> </mtd> <mtd> <mn>100</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0.1</mn> </mtd> <mtd> <mn>0.01</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Decimal example digit</mtext> </mrow> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c26c9b64098769a9ffd3549c273a753b6a922cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:66.242ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{l|rrrrrrr}{\text{Position}}&3&2&1&0&-1&-2&\cdots \\\hline {\text{Weight}}&b^{3}&b^{2}&b^{1}&b^{0}&b^{-1}&b^{-2}&\cdots \\{\text{Digit}}&a_{3}&a_{2}&a_{1}&a_{0}&c_{1}&c_{2}&\cdots \\\hline {\text{Decimal example weight}}&1000&100&10&1&0.1&0.01&\cdots \\{\text{Decimal example digit}}&4&3&2&7&0&0&\cdots \end{array}}}"></span></dd></dl> <p>A number has a terminating or repeating expansion <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Rational_number" title="Rational number">rational</a>; this does not depend on the base. A number that terminates in one base may repeat in another (thus <span class="texhtml">0.3<sub>10</sub> = 0.0100110011001...<sub>2</sub></span>). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, <span class="texhtml"><a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> = 3.1415926...<sub>10</sub></span> can be written as the aperiodic 11.001001000011111...<sub>2</sub>. </p><p>Putting <a href="/wiki/Overline" title="Overline">overscores</a>, <span style="text-decoration:overline;"><i>n</i></span>, or dots, <i>ṅ</i>, above the common digits is a convention used to represent repeating rational expansions. Thus: </p> <dl><dd>14/11 = 1.272727272727... = 1.<span style="text-decoration:overline;">27</span>   or   321.3217878787878... = 321.321<span style="text-decoration:overline;">78</span>.</dd></dl> <p>If <i>b</i> = <i>p</i> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>, one can define base-<i>p</i> numerals whose expansion to the left never stops; these are called the <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>. </p><p>It is also possible to define a variation of base <i>b</i> in which digits may be positive or negative; this is called a <a href="/wiki/Signed-digit_representation" title="Signed-digit representation">signed-digit representation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalized_variable-length_integers">Generalized variable-length integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=5" title="Edit section: Generalized variable-length integers"><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/Punycode" title="Punycode">Punycode</a></div> <p>More general is using a <a href="/wiki/Mixed_radix" title="Mixed radix">mixed radix</a> notation (here written <a href="/wiki/Endianness" title="Endianness">little-endian</a>) like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}a_{1}a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}a_{1}a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594f4bccf62e3e1214c2720d00534ba007e5f03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.852ex; height:2.009ex;" alt="{\displaystyle a_{0}a_{1}a_{2}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5eb27e8a6336e82759411076ccfacfd5bdc040a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.688ex; height:2.509ex;" alt="{\displaystyle a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}}"></span>, etc. </p><p>This is used in <a href="/wiki/Punycode" title="Punycode">Punycode</a>, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26–35 respectively. There are also so-called threshold values (<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 t_{0},t_{1},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0},t_{1},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ea6d4b602a6146c0b7ec04bf9d459c1b7541fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.579ex; height:2.343ex;" alt="{\displaystyle t_{0},t_{1},\ldots }"></span>) which are fixed for every position in the number. A digit <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> (in a given position in the number) that is lower than its corresponding threshold value <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 t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> means that it is the most-significant digit, hence in the string this is the end of the number, and the next symbol (if present) is the least-significant digit of the next number. </p><p>For example, if the threshold value for the first digit is <i>b</i> (i.e. 1) then <i>a</i> (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit, first-digit range is only b–9 (i.e. 1–35), therefore the weight <i>b</i><sub>1</sub> is 35 instead of 36. More generally, if <i>t<sub>n</sub></i> is the threshold for the <i>n</i>-th digit, it is easy to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n+1}=36-t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>36</mn> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n+1}=36-t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486a14aabdc64357bf42d01337b677ca5fb5ec76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.638ex; height:2.509ex;" alt="{\displaystyle b_{n+1}=36-t_{n}}"></span>. Suppose the threshold values for the second and third digits are <i>c</i> (i.e. 2), then the second-digit range is a–b (i.e. 0–1) with the second digit being most significant, while the range is c–9 (i.e. 2–35) in the presence of a third digit. Generally, for any <i>n</i>, the weight of the (<i>n</i> + 1)-th digit is the weight of the previous one times (36 − threshold of the <i>n</i>-th digit). So the weight of the second symbol is <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 36-t_{0}=35}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>35</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36-t_{0}=35}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2608d79127bafa4f5f85fab420769cceb5e388c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.483ex; height:2.509ex;" alt="{\displaystyle 36-t_{0}=35}"></span>. And the weight of the third symbol is <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 35(36-t_{1})=35\cdot 34=1190}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>35</mn> <mo stretchy="false">(</mo> <mn>36</mn> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>35</mn> <mo>⋅</mo> <mn>34</mn> <mo>=</mo> <mn>1190</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 35(36-t_{1})=35\cdot 34=1190}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17f8f112c251fe5804f3bbac589cf2a544d3a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.369ex; height:2.843ex;" alt="{\displaystyle 35(36-t_{1})=35\cdot 34=1190}"></span>. </p><p>So we have the following sequence of the numbers with at most 3 digits: </p><p><i>a</i> (0), <i>ba</i> (1), <i>ca</i> (2), ..., 9<i>a</i> (35), <i>bb</i> (36), <i>cb</i> (37), ..., 9<i>b</i> (70), <i>bca</i> (71), ..., 99<i>a</i> (1260), <i>bcb</i> (1261), ..., 99<i>b</i> (2450). </p><p>Unlike a regular <i>n</i>-based numeral system, there are numbers like 9<i>b</i> where 9 and <i>b</i> each represent 35; yet the representation is unique because <i>ac</i> and <i>aca</i> are not allowed – the first <i>a</i> would terminate each of these numbers. </p><p>The flexibility in choosing threshold values allows optimization for number of digits depending on the frequency of occurrence of numbers of various sizes. </p><p>The case with all threshold values equal to 1 corresponds to <a href="/wiki/Bijective_numeration" title="Bijective numeration">bijective numeration</a>, where the zeros correspond to separators of numbers with digits which are non-zero. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=6" 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: 22em;"> <ul><li><a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></li> <li><a href="/wiki/Computer_number_format" title="Computer number format">Computer number formats</a></li> <li><a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard positional numeral systems</a></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History of ancient numeral systems</a></li> <li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">History of numbers</a></li> <li><a href="/wiki/List_of_numeral_system_topics" title="List of numeral system topics">List of numeral system topics</a></li> <li><a href="/wiki/Numeral_(linguistics)" title="Numeral (linguistics)">Number names</a></li> <li><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a></li> <li><a href="/wiki/Residue_numeral_system" class="mw-redirect" title="Residue numeral system">Residue numeral system</a></li> <li><a href="/wiki/Long_and_short_scales" title="Long and short scales">Long and short scales</a></li> <li><a href="/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li> <li><a href="/wiki/-yllion" title="-yllion">-yllion</a></li> <li><a href="/wiki/Numerical_cognition" title="Numerical cognition">Numerical cognition</a></li> <li><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number system</a></li></ul></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=Numeral_system&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-O'Connor_and_Robertson-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-O'Connor_and_Robertson_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-O'Connor_and_Robertson_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">O'Connor, J. J. and Robertson, E. F. <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_numerals.html">Arabic Numerals</a>. January 2001. Retrieved on 2007-02-20.</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"><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="CITEREFBill_Casselman2007" class="citation web cs1"><a href="/wiki/Bill_Casselman_(mathematician)" class="mw-redirect" title="Bill Casselman (mathematician)">Bill Casselman</a> (February 2007). <a rel="nofollow" class="external text" href="https://www.ams.org/featurecolumn/archive/india-zero.html">"All for Nought"</a>. <i>Feature Column</i>. AMS.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Feature+Column&rft.atitle=All+for+Nought&rft.date=2007-02&rft.au=Bill+Casselman&rft_id=https%3A%2F%2Fwww.ams.org%2Ffeaturecolumn%2Farchive%2Findia-zero.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradley" class="citation web cs1">Bradley, Jeremy. <a rel="nofollow" class="external text" href="https://www.theclassroom.com/how-to-identify-numbers-on-brass-from-india-12082499.html">"How Arabic Numbers Were Invented"</a>. <i>www.theclassroom.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.theclassroom.com&rft.atitle=How+Arabic+Numbers+Were+Invented&rft.aulast=Bradley&rft.aufirst=Jeremy&rft_id=https%3A%2F%2Fwww.theclassroom.com%2Fhow-to-identify-numbers-on-brass-from-india-12082499.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2004" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> (January 2004), <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Chinese_numerals.html">"Chinese numerals"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chinese+numerals&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.date=2004-01&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FChinese_numerals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span> </li> <li id="cite_note-Crossley-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Crossley_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShenCrossleyLun1999" class="citation book cs1">Shen Kanshen Crossley, John N.; Lun, Anthony W.-C. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eiTJHRGTG6YC"><i>The Nine Chapters on the Mathematical Art: Companion and Commentary</i></a>. Oxford University Press. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853936-0" title="Special:BookSources/978-0-19-853936-0"><bdi>978-0-19-853936-0</bdi></a>. <q>zero was regarded as a number in India ... whereas the Chinese employed a vacant position</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nine+Chapters+on+the+Mathematical+Art%3A+Companion+and+Commentary&rft.pages=35&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-853936-0&rft.aulast=Shen&rft.aufirst=Kangshen&rft.au=Crossley%2C+John+N.&rft.au=Lun%2C+Anthony+W.-C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeiTJHRGTG6YC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span> </li> <li id="cite_note-Qin-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Qin_6-0">^</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://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf">"Mathematics in the Near and Far East"</a> <span class="cs1-format">(PDF)</span>. <i>grmath4.phpnet.us</i>. p. 262. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131104120005/http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 4 November 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">7 June</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=grmath4.phpnet.us&rft.atitle=Mathematics+in+the+Near+and+Far+East&rft.pages=262&rft_id=http%3A%2F%2Fgrmath4.phpnet.us%2Fistoria%2Fthe_history_of%2520math_greece%2Fthe_history_of%2520math_greece_3-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></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="CITEREFMartzloff2007" class="citation book cs1">Martzloff, Jean-Claude (2007). <i>A History of Chinese Mathematics</i>. Translated by Wilson, Stephen S. Springer. p. 208. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33783-6" title="Special:BookSources/978-3-540-33783-6"><bdi>978-3-540-33783-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Chinese+Mathematics&rft.pages=208&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-33783-6&rft.aulast=Martzloff&rft.aufirst=Jean-Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Eugene_SmithLouis_Charles_Karpinski1911" class="citation book cs1">David Eugene Smith; Louis Charles Karpinski (1911). <a rel="nofollow" class="external text" href="https://archive.org/details/hinduarabicnume05karpgoog"><i>The Hindu–Arabic numerals</i></a>. Ginn and Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Hindu%E2%80%93Arabic+numerals&rft.pub=Ginn+and+Company&rft.date=1911&rft.au=David+Eugene+Smith&rft.au=Louis+Charles+Karpinski&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhinduarabicnume05karpgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" 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 id="CITEREFChowdhury" class="citation book cs1">Chowdhury, Arnab. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WXn-mT3K6dgC&q=Arithmetic+is+much+easier+in+positional+systems+than+in+the+earlier+additive+ones;+furthermore,+additive+systems+need+a+large+number+of+different+symbols+for+the+different+powers+of+10;+a+positional+system+needs+only+ten+different+symbols+(assuming+that+it+uses+base+10).&pg=PA2"><i>Design of an Efficient Multiplier using DBNS</i></a>. GIAP Journals. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-93-83006-18-2" title="Special:BookSources/978-93-83006-18-2"><bdi>978-93-83006-18-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+of+an+Efficient+Multiplier+using+DBNS&rft.pub=GIAP+Journals&rft.isbn=978-93-83006-18-2&rft.aulast=Chowdhury&rft.aufirst=Arnab&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWXn-mT3K6dgC%26q%3DArithmetic%2Bis%2Bmuch%2Beasier%2Bin%2Bpositional%2Bsystems%2Bthan%2Bin%2Bthe%2Bearlier%2Badditive%2Bones%3B%2Bfurthermore%2C%2Badditive%2Bsystems%2Bneed%2Ba%2Blarge%2Bnumber%2Bof%2Bdifferent%2Bsymbols%2Bfor%2Bthe%2Bdifferent%2Bpowers%2Bof%2B10%3B%2Ba%2Bpositional%2Bsystem%2Bneeds%2Bonly%2Bten%2Bdifferent%2Bsymbols%2B%28assuming%2Bthat%2Bit%2Buses%2Bbase%2B10%29.%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" 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"> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=8" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Georges Ifrah. <i>The Universal History of Numbers : From Prehistory to the Invention of the Computer</i>, Wiley, 1999. <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/0-471-37568-3" title="Special:BookSources/0-471-37568-3">0-471-37568-3</a>.</li> <li><a href="/wiki/Donald_Knuth" title="Donald Knuth">D. Knuth</a>. <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>. Volume 2, 3rd Ed. <a href="/wiki/Addison%E2%80%93Wesley" class="mw-redirect" title="Addison–Wesley">Addison–Wesley</a>. pp. 194–213, "Positional Number Systems".</li> <li><a href="/wiki/A.L._Kroeber" class="mw-redirect" title="A.L. Kroeber">A.L. Kroeber</a> (Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919)</li> <li>J.P. Mallory; D.Q. Adams, <i>Encyclopedia of Indo-European Culture</i>, Fitzroy Dearborn Publishers, London and Chicago, 1997.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHans_J._NissenPeter_DamerowRobert_K._Englund1993" class="citation book cs1">Hans J. Nissen; Peter Damerow; Robert K. Englund (1993). <i>Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East</i>. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-58659-5" title="Special:BookSources/978-0-226-58659-5"><bdi>978-0-226-58659-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Archaic+Bookkeeping%3A+Early+Writing+and+Techniques+of+Economic+Administration+in+the+Ancient+Near+East&rft.pub=University+of+Chicago+Press&rft.date=1993&rft.isbn=978-0-226-58659-5&rft.au=Hans+J.+Nissen&rft.au=Peter+Damerow&rft.au=Robert+K.+Englund&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmandt-Besserat1996" class="citation book cs1"><a href="/wiki/Denise_Schmandt-Besserat" title="Denise Schmandt-Besserat">Schmandt-Besserat, Denise</a> (1996). <i>How Writing Came About</i>. <a href="/wiki/University_of_Texas_Press" title="University of Texas Press">University of Texas Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-292-77704-0" title="Special:BookSources/978-0-292-77704-0"><bdi>978-0-292-77704-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Writing+Came+About&rft.pub=University+of+Texas+Press&rft.date=1996&rft.isbn=978-0-292-77704-0&rft.aulast=Schmandt-Besserat&rft.aufirst=Denise&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaslavsky1999" class="citation book cs1"><a href="/wiki/Claudia_Zaslavsky" title="Claudia Zaslavsky">Zaslavsky, Claudia</a> (1999). <i>Africa counts: number and pattern in African cultures</i>. Chicago Review Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55652-350-2" title="Special:BookSources/978-1-55652-350-2"><bdi>978-1-55652-350-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Africa+counts%3A+number+and+pattern+in+African+cultures&rft.pub=Chicago+Review+Press&rft.date=1999&rft.isbn=978-1-55652-350-2&rft.aulast=Zaslavsky&rft.aufirst=Claudia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANumeral+system" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeral_system&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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