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class="vector-toc-link" href="#In_computers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>In computers</span> </div> </a> <ul id="toc-In_computers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_aviation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_aviation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>In aviation</span> </div> </a> <ul id="toc-In_aviation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conversion_between_bases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conversion_between_bases"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Conversion between bases</span> </div> </a> <button aria-controls="toc-Conversion_between_bases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Conversion between bases subsection</span> </button> <ul id="toc-Conversion_between_bases-sublist" class="vector-toc-list"> <li id="toc-Decimal_to_octal_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decimal_to_octal_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Decimal to octal conversion</span> </div> </a> <ul id="toc-Decimal_to_octal_conversion-sublist" class="vector-toc-list"> <li id="toc-Method_of_successive_Euclidean_division_by_8" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_successive_Euclidean_division_by_8"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Method of successive Euclidean division by 8</span> </div> </a> <ul id="toc-Method_of_successive_Euclidean_division_by_8-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_of_successive_multiplication_by_8" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_successive_multiplication_by_8"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Method of successive multiplication by 8</span> </div> </a> <ul id="toc-Method_of_successive_multiplication_by_8-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_of_successive_duplication" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_successive_duplication"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Method of successive duplication</span> </div> </a> <ul id="toc-Method_of_successive_duplication-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Octal_to_decimal_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Octal_to_decimal_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Octal to decimal conversion</span> </div> </a> <ul id="toc-Octal_to_decimal_conversion-sublist" class="vector-toc-list"> <li id="toc-Method_of_successive_duplication_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_successive_duplication_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Method of successive duplication</span> </div> </a> <ul id="toc-Method_of_successive_duplication_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Octal_to_binary_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Octal_to_binary_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Octal to binary conversion</span> </div> </a> <ul id="toc-Octal_to_binary_conversion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Binary_to_octal_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_to_octal_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Binary to octal conversion</span> </div> </a> <ul id="toc-Binary_to_octal_conversion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Octal_to_hexadecimal_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Octal_to_hexadecimal_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Octal to hexadecimal conversion</span> </div> </a> <ul id="toc-Octal_to_hexadecimal_conversion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hexadecimal_to_octal_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hexadecimal_to_octal_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Hexadecimal to octal conversion</span> </div> </a> <ul id="toc-Hexadecimal_to_octal_conversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Real_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Real numbers</span> </div> </a> <button aria-controls="toc-Real_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Real numbers subsection</span> </button> <ul id="toc-Real_numbers-sublist" class="vector-toc-list"> <li id="toc-Fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fractions</span> </div> </a> <ul id="toc-Fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irrational_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irrational_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Irrational numbers</span> </div> </a> <ul id="toc-Irrational_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</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">5</span> <span>References</span> </div> </a> <ul id="toc-References-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">6</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" title="Table of Contents" > <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">Octal</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 64 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-64" 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">64 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%B9%D8%AF_%D8%AB%D9%85%D8%A7%D9%86%D9%8A" title="نظام عد ثماني – Arabic" lang="ar" hreflang="ar" data-title="نظام عد ثماني" 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/S%C9%99kkizlik_say_sistemi" title="Səkkizlik say sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Səkkizlik say sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%B7%E0%A7%8D%E0%A6%9F%E0%A6%95_%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="অষ্টক সংখ্যা পদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="অষ্টক সংখ্যা পদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Pat-ch%C3%ACn-hoat" title="Pat-chìn-hoat – Minnan" lang="nan" hreflang="nan" data-title="Pat-chìn-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%81%D0%BC%D0%B8%D1%87%D0%BD%D0%B0_%D0%B1%D1%80%D0%BE%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Осмична бройна система – Bulgarian" lang="bg" hreflang="bg" data-title="Осмична 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/Oktalni_sistem" title="Oktalni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Oktalni 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_octal" title="Sistema octal – Catalan" lang="ca" hreflang="ca" data-title="Sistema octal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Osmi%C4%8Dkov%C3%A1_soustava" title="Osmičková soustava – Czech" lang="cs" hreflang="cs" data-title="Osmičková soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Oktale_talsystem" title="Oktale talsystem – Danish" lang="da" hreflang="da" data-title="Oktale 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/Oktalsystem" title="Oktalsystem – German" lang="de" hreflang="de" data-title="Oktalsystem" 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/Kaheksands%C3%BCsteem" title="Kaheksandsüsteem – Estonian" lang="et" hreflang="et" data-title="Kaheksandsüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BA%CF%84%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B1%CF%81%CE%AF%CE%B8%CE%BC%CE%B7%CF%83%CE%B7%CF%82" title="Οκταδικό σύστημα αρίθμησης – Greek" lang="el" hreflang="el" data-title="Οκταδικό σύστημα αρίθμησης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_octal" title="Sistema octal – Spanish" lang="es" hreflang="es" data-title="Sistema octal" 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/Okuma_nombrosistemo" title="Okuma nombrosistemo – Esperanto" lang="eo" hreflang="eo" data-title="Okuma 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_zortzitar" title="Zenbaki-sistema zortzitar – Basque" lang="eu" hreflang="eu" data-title="Zenbaki-sistema zortzitar" 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%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%BE%D8%A7%DB%8C%D9%87_%DB%B8" title="دستگاه اعداد پایه ۸ – Persian" lang="fa" hreflang="fa" data-title="دستگاه اعداد پایه ۸" 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_octal" title="Système octal – French" lang="fr" hreflang="fr" data-title="Système octal" 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/C%C3%B3digo_octal" title="Código octal – Galician" lang="gl" hreflang="gl" data-title="Código octal" 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/%ED%8C%94%EC%A7%84%EB%B2%95" title="팔진법 – Korean" lang="ko" hreflang="ko" data-title="팔진법" 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%B8%D6%82%D5%A9%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Հաշվարկման ութական համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Հաշվարկման ութական համակարգ" 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%85%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BE%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A5%80" title="अष्टाधारी – Hindi" lang="hi" hreflang="hi" data-title="अष्टाधारी" 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/Oktalni_brojevni_sustav" title="Oktalni brojevni sustav – Croatian" lang="hr" hreflang="hr" data-title="Oktalni 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/Oktala_nombrosistemo" title="Oktala nombrosistemo – Ido" lang="io" hreflang="io" data-title="Oktala 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/Oktal" title="Oktal – Indonesian" lang="id" hreflang="id" data-title="Oktal" 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/%C3%81ttundakerfi" title="Áttundakerfi – Icelandic" lang="is" hreflang="is" data-title="Áttundakerfi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_numerico_ottale" title="Sistema numerico ottale – Italian" lang="it" hreflang="it" data-title="Sistema numerico ottale" 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%91%D7%A1%D7%99%D7%A1_%D7%90%D7%95%D7%A7%D7%98%D7%9C%D7%99" title="בסיס אוקטלי – Hebrew" lang="he" hreflang="he" data-title="בסיס אוקטלי" 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_oktal" title="Sistem wilangan oktal – Javanese" lang="jv" hreflang="jv" data-title="Sistem wilangan oktal" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_oktal" title="Sistèm oktal – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm oktal" 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-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_numericum_octale" title="Systema numericum octale – Latin" lang="la" hreflang="la" data-title="Systema numericum octale" 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/Okt%C4%81l%C4%81_skait%C4%AB%C5%A1anas_sist%C4%93ma" title="Oktālā skaitīšanas sistēma – Latvian" lang="lv" hreflang="lv" data-title="Oktālā skaitīšanas sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/A%C5%A1tuntain%C4%97_skai%C4%8Diavimo_sistema" title="Aštuntainė skaičiavimo sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Aštuntainė skaičiavimo sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Nyolcas_sz%C3%A1mrendszer" title="Nyolcas számrendszer – Hungarian" lang="hu" hreflang="hu" data-title="Nyolcas 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%9E%D1%81%D0%BC%D0%B5%D1%80%D0%B5%D1%87%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Осмеречен броен систем – Macedonian" lang="mk" hreflang="mk" data-title="Осмеречен 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-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sistema_otal" title="Sistema otal – Mirandese" lang="mwl" hreflang="mwl" data-title="Sistema otal" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Octaal_talstelsel" title="Octaal talstelsel – Dutch" lang="nl" hreflang="nl" data-title="Octaal 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%85%AB%E9%80%B2%E6%B3%95" title="八進法 – Japanese" lang="ja" hreflang="ja" data-title="八進法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/%C3%85ttetallsystemet" title="Åttetallsystemet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Åttetallsystemet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/%C3%85ttetalssystemet" title="Åttetalssystemet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Åttetalssystemet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sakkizlik_sanoq_sistemasi" title="Sakkizlik sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Sakkizlik 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C3%93semkowy_system_liczbowy" title="Ósemkowy system liczbowy – Polish" lang="pl" hreflang="pl" data-title="Ósemkowy 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_octal" title="Sistema octal – Portuguese" lang="pt" hreflang="pt" data-title="Sistema octal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_octal" title="Sistem octal – Romanian" lang="ro" hreflang="ro" data-title="Sistem octal" 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%92%D0%BE%D1%81%D1%8C%D0%BC%D0%B5%D1%80%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Восьмеричная система счисления – Russian" lang="ru" hreflang="ru" data-title="Восьмеричная 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-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D8%A2%DA%A9%D9%B9%D9%84_%D9%86%D9%85%D8%A8%D8%B1_%D8%B3%D8%B3%D9%B9%D9%85" title="آکٹل نمبر سسٹم – Saraiki" lang="skr" hreflang="skr" data-title="آکٹل نمبر سسٹم" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target"><span>سرائیکی</span></a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Letlase_la_seswai" title="Letlase la seswai – Northern Sotho" lang="nso" hreflang="nso" data-title="Letlase la seswai" 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-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_oktal" title="Sistemi oktal – Albanian" lang="sq" hreflang="sq" data-title="Sistemi oktal" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Octal" title="Octal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Octal" 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-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D9%8E%D9%BA%D9%86_%D9%88%D8%A7%D8%B1%D9%8A_%D9%BE%D9%8A%DA%99%D9%87%D9%87_%D8%AC%D9%88_%D9%86%D8%B8%D8%A7%D9%85" title="اَٺن واري پيڙهه جو نظام – Sindhi" lang="sd" hreflang="sd" data-title="اَٺن واري پيڙهه جو نظام" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Osmi%C4%8Dkov%C3%A1_s%C3%BAstava" title="Osmičková sústava – Slovak" lang="sk" hreflang="sk" data-title="Osmičková sústava" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%BA%D1%82%D0%B0%D0%BB%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Октални систем – Serbian" lang="sr" hreflang="sr" data-title="Октални 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/Oktalni_sistem" title="Oktalni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Oktalni 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/Oktaalij%C3%A4rjestelm%C3%A4" title="Oktaalijärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Oktaalijärjestelmä" 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/Oktala_talsystemet" title="Oktala talsystemet – Swedish" lang="sv" hreflang="sv" data-title="Oktala talsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%A3%E0%AF%86%E0%AE%A3%E0%AF%8D" title="எண்ணெண் – Tamil" lang="ta" hreflang="ta" data-title="எண்ணெண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B9%81%E0%B8%9B%E0%B8%94" title="เลขฐานแปด – Thai" lang="th" hreflang="th" data-title="เลขฐานแปด" 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/Sekizli_say%C4%B1_sistemi" title="Sekizli sayı sistemi – Turkish" lang="tr" hreflang="tr" data-title="Sekizli sayı sistemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D1%81%D1%96%D0%BC%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Вісімкова система числення – Ukrainian" lang="uk" hreflang="uk" data-title="Вісімкова 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%A7%D8%B3%D8%A7%D8%B3_%D8%A2%D9%B9%DA%BE_%DA%A9%D8%A7_%D9%86%D8%B8%D8%A7%D9%85" title="اساس آٹھ کا نظام – Urdu" lang="ur" hreflang="ur" data-title="اساس آٹھ کا 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_b%C3%A1t_ph%C3%A2n" title="Hệ bát phân – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ bát phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%85%AB%E9%80%B2%E5%88%B6" title="八進制 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="八進制" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%85%AB%E8%BF%9B%E5%88%B6" title="八进制 – Wu" lang="wuu" hreflang="wuu" data-title="八进制" 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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%AB%E9%80%B2%E5%88%B6" title="八進制 – Cantonese" lang="yue" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Base-8 numeral system</div> <table class="sortable" border="0" cellspacing="0" cellpadding="0" style="color:black; text-align:center; border:2px solid black; float:right;"> <caption><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a>, <a href="/wiki/Bit" title="Bit">bits</a> and <a href="/wiki/Gray_code" title="Gray code">Gray code</a> </caption> <tbody><tr style="background:#cc9;"> <th><a href="/wiki/Hexadecimal" title="Hexadecimal">hex</a></th> <th><a href="/wiki/Decimal" title="Decimal">dec</a></th> <th><a class="mw-selflink selflink">oct</a></th> <th class="unsortable" style="background:#c9c;">3</th> <th class="unsortable" style="background:#c9c;">2</th> <th class="unsortable" style="background:#c9c;">1</th> <th class="unsortable" style="background:#c9c;">0</th> <th><a href="/wiki/Gray_code" title="Gray code">step</a> </th></tr> <tr style="background:#c0c0ff;"> <td><b>0</b><sub>hex</sub></td> <td><span style="display:none">0</span>0<sub>dec</sub></td> <td><span style="display:none">0</span>0<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">g</span>0 </td></tr> <tr style="background:white;"> <td><b>1</b><sub>hex</sub></td> <td><span style="display:none">0</span>1<sub>dec</sub></td> <td><span style="display:none">0</span>1<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">h</span>1 </td></tr> <tr style="background:white;"> <td><b>2</b><sub>hex</sub></td> <td><span style="display:none">0</span>2<sub>dec</sub></td> <td><span style="display:none">0</span>2<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">j</span>3 </td></tr> <tr style="background:#e1e1ff;"> <td><b>3</b><sub>hex</sub></td> <td><span style="display:none">0</span>3<sub>dec</sub></td> <td><span style="display:none">0</span>3<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">i</span>2 </td></tr> <tr style="background:white;; border-top:2px solid black;"> <td><b>4</b><sub>hex</sub></td> <td><span style="display:none">0</span>4<sub>dec</sub></td> <td><span style="display:none">0</span>4<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">n</span>7 </td></tr> <tr style="background:#e1e1ff;"> <td><b>5</b><sub>hex</sub></td> <td><span style="display:none">0</span>5<sub>dec</sub></td> <td><span style="display:none">0</span>5<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">m</span>6 </td></tr> <tr style="background:#e1ffe1;"> <td><b>6</b><sub>hex</sub></td> <td><span style="display:none">0</span>6<sub>dec</sub></td> <td><span style="display:none">0</span>6<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">k</span>4 </td></tr> <tr style="background:white;"> <td><b>7</b><sub>hex</sub></td> <td><span style="display:none">0</span>7<sub>dec</sub></td> <td><span style="display:none">0</span>7<sub>oct</sub></td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">l</span>5 </td></tr> <tr style="background:white;; border-top:2px solid black;"> <td><b>8</b><sub>hex</sub></td> <td><span style="display:none">0</span>8<sub>dec</sub></td> <td>10<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">v</span>F </td></tr> <tr style="background:#e1ffe1;"> <td><b>9</b><sub>hex</sub></td> <td><span style="display:none">0</span>9<sub>dec</sub></td> <td>11<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">u</span>E </td></tr> <tr style="background:#e1e1ff;"> <td><b>A</b><sub>hex</sub></td> <td>10<sub>dec</sub></td> <td>12<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">s</span>C </td></tr> <tr style="background:white;"> <td><b>B</b><sub>hex</sub></td> <td>11<sub>dec</sub></td> <td>13<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">t</span>D </td></tr> <tr style="background:#e1e1ff;; border-top:2px solid black;"> <td><b>C</b><sub>hex</sub></td> <td>12<sub>dec</sub></td> <td>14<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:white;width:24px">0</td> <td><span style="display:none">o</span>8 </td></tr> <tr style="background:white;"> <td><b>D</b><sub>hex</sub></td> <td>13<sub>dec</sub></td> <td>15<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td style="background:white;width:24px">0</td> <td style="background:red;width:24px">1</td> <td><span style="display:none">p</span>9 </td></tr> <tr style="background:white;"> <td><b>E</b><sub>hex</sub></td> <td>14<sub>dec</sub></td> <td>16<sub>oct</sub></td> <td style="background:red;width:24px">1</td> <td style="background:red;width:24px">1</td> <td 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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/Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li> <li><a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></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 class="mw-selflink selflink">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" class="mw-redirect" title="Prehistoric counting">Prehistoric counting</a></li> <li><a href="/wiki/Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li> <li><a href="/wiki/Roman_numerals" title="Roman numerals">Roman</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li></ul> <hr /> <dl><dt><a href="/wiki/Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl> <ul><li><a href="/wiki/Abjad_numerals" title="Abjad numerals">Abjad</a></li> <li><a href="/wiki/Armenian_numerals" title="Armenian numerals">Armenian</a></li> <li><a href="/wiki/Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a> <ul><li><a href="/wiki/Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li> <li><a href="/wiki/Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li> <li><a href="/wiki/Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li> <li><a href="/wiki/Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li> <li><a href="/wiki/Georgian_numerals" title="Georgian numerals">Georgian</a></li> <li><a href="/wiki/Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek</a></li> <li><a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numeral_systems" title="Template:Numeral systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numeral_systems" title="Template talk:Numeral systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numeral_systems" title="Special:EditPage/Template:Numeral systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Octal</b> (<b>base 8</b>) is a <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> with <a href="/wiki/8_(number)" class="mw-redirect" title="8 (number)">eight</a> as the <a href="/wiki/Radix" title="Radix">base</a>. </p><p>In the decimal system, each place is a <a href="/wiki/Power_of_ten" class="mw-redirect" title="Power of ten">power of ten</a>. For 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 \mathbf {74} _{10}=\mathbf {7} \times 10^{1}+\mathbf {4} \times 10^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">74</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">7</mn> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">4</mn> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {74} _{10}=\mathbf {7} \times 10^{1}+\mathbf {4} \times 10^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79337e370dac156042f618e518bb41e369d55c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.601ex; height:3.009ex;" alt="{\displaystyle \mathbf {74} _{10}=\mathbf {7} \times 10^{1}+\mathbf {4} \times 10^{0}}" /></span></dd></dl> <p>In the octal system, each place is a power of eight. For 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 \mathbf {112} _{8}=\mathbf {1} \times 8^{2}+\mathbf {1} \times 8^{1}+\mathbf {2} \times 8^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">112</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>×</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>×</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mo>×</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {112} _{8}=\mathbf {1} \times 8^{2}+\mathbf {1} \times 8^{1}+\mathbf {2} \times 8^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fac38a4dc6bfc166a437e00b9ec51f1714d1f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.025ex; height:3.009ex;" alt="{\displaystyle \mathbf {112} _{8}=\mathbf {1} \times 8^{2}+\mathbf {1} \times 8^{1}+\mathbf {2} \times 8^{0}}" /></span></dd></dl> <p>By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to <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 64+8+2=74}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>64</mn> <mo>+</mo> <mn>8</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>74</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 64+8+2=74}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf0711e3868eba9af35f29c5995a8a85d843bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.754ex; height:2.343ex;" alt="{\displaystyle 64+8+2=74}" /></span> in decimal. </p><p>Octal numerals can be easily converted from <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> representations (similar to a <a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">quaternary numeral system</a>) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: <span class="nowrap">(00)1 001 010</span>, corresponding to the octal digits <span class="nowrap">1 1 2</span>, yielding the octal representation 112. </p> <table class="wikitable" style="float:right; text-align:center"> <caption>The octal <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a> </caption> <tbody><tr> <td>×</td> <td><b>1</b></td> <td><b>2</b></td> <td><b>3</b></td> <td><b>4</b></td> <td><b>5</b></td> <td><b>6</b></td> <td><b>7</b></td> <td><b>10</b> </td></tr> <tr> <td><b>1</b></td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>10 </td></tr> <tr> <td><b>2</b></td> <td>2</td> <td>4</td> <td>6</td> <td>10</td> <td>12</td> <td>14</td> <td>16</td> <td>20 </td></tr> <tr> <td><b>3</b></td> <td>3</td> <td>6</td> <td>11</td> <td>14</td> <td>17</td> <td>22</td> <td>25</td> <td>30 </td></tr> <tr> <td><b>4</b></td> <td>4</td> <td>10</td> <td>14</td> <td>20</td> <td>24</td> <td>30</td> <td>34</td> <td>40 </td></tr> <tr> <td><b>5</b></td> <td>5</td> <td>12</td> <td>17</td> <td>24</td> <td>31</td> <td>36</td> <td>43</td> <td>50 </td></tr> <tr> <td><b>6</b></td> <td>6</td> <td>14</td> <td>22</td> <td>30</td> <td>36</td> <td>44</td> <td>52</td> <td>60 </td></tr> <tr> <td><b>7</b></td> <td>7</td> <td>16</td> <td>25</td> <td>34</td> <td>43</td> <td>52</td> <td>61</td> <td>70 </td></tr> <tr> <td><b>10</b></td> <td>10</td> <td>20</td> <td>30</td> <td>40</td> <td>50</td> <td>60</td> <td>70</td> <td>100 </td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=1" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_China">In China</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=2" title="Edit section: In China"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bagua-name-earlier.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Bagua-name-earlier.svg/250px-Bagua-name-earlier.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Bagua-name-earlier.svg/375px-Bagua-name-earlier.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Bagua-name-earlier.svg/500px-Bagua-name-earlier.svg.png 2x" data-file-width="547" data-file-height="547" /></a><figcaption><a href="/wiki/Fuxi" title="Fuxi">Fuxi</a>'s "Earlier Heaven" Arrangement of the Eight Trigrams</figcaption></figure> <p>The eight <a href="/wiki/Bagua" title="Bagua">bagua</a> or trigrams of the <a href="/wiki/I_Ching" title="I Ching">I Ching</a> correspond to octal digits: </p> <ul><li>0 = ☷, 1 = ☳, 2 = ☵, 3 = ☱,</li> <li>4 = ☶, 5 = ☲, 6 = ☴, 7 = ☰.</li></ul> <p><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> made the connection between trigrams, hexagrams and binary numbers in 1703.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="By_Native_Americans">By Native Americans</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=3" title="Edit section: By Native Americans"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Yuki_language" title="Yuki language">Yuki language</a> in <a href="/wiki/California" title="California">California</a> has an octal system because the speakers count using the spaces between their fingers rather than the fingers themselves.<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></li> <li>The <a href="/wiki/Pame_language" class="mw-redirect" title="Pame language">Pamean languages</a> in <a href="/wiki/Mexico" title="Mexico">Mexico</a> also have an octal system, because their speakers count on the knuckles of a closed fist.<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></li></ul> <div class="mw-heading mw-heading3"><h3 id="By_Europeans">By Europeans</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=4" title="Edit section: By Europeans"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>It has been suggested that the reconstructed <a href="/wiki/Proto-Indo-European" class="mw-redirect" title="Proto-Indo-European">Proto-Indo-European (PIE)</a> word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal number system, though the evidence supporting this is slim.<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></li> <li>In 1668, <a href="/wiki/John_Wilkins" title="John Wilkins">John Wilkins</a> in <i><a href="/wiki/An_Essay_towards_a_Real_Character,_and_a_Philosophical_Language" class="mw-redirect" title="An Essay towards a Real Character, and a Philosophical Language">An Essay towards a Real Character, and a Philosophical Language</a></i> proposed use of base 8 instead of 10 "because the way of Dichotomy or Bipartition being the most natural and easie kind of Division, that Number is capable of this down to an Unite".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>In 1716, King <a href="/wiki/Charles_XII_of_Sweden" title="Charles XII of Sweden">Charles XII of Sweden</a> asked <a href="/wiki/Emanuel_Swedenborg" title="Emanuel Swedenborg">Emanuel Swedenborg</a> to elaborate a number system based on 64 instead of 10. Swedenborg argued, however, that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as the base. In 1718 Swedenborg wrote (but did not publish) a manuscript: "<i>En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10</i>" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 1–7 are there denoted by the consonants l, s, n, m, t, f, u (v) and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>Writing under the pseudonym "Hirossa Ap-Iccim" in <i><a href="/wiki/The_Gentleman%27s_Magazine" title="The Gentleman's Magazine">The Gentleman's Magazine</a></i>, (London) July 1745, <a href="/wiki/Hugh_Jones_(reverend)" class="mw-redirect" title="Hugh Jones (reverend)">Hugh Jones</a> proposed an octal system for British coins, weights and measures. "Whereas reason and convenience indicate to us an uniform standard for all quantities; which I shall call the <i>Georgian standard</i>; and that is only to divide every integer in each <i>species</i> into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho' all nations count universally by <i>tens</i> (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision <i>ten</i> is uncapable...." In a later treatise on <a href="/wiki/Hugh_Jones_(reverend)#Publications" class="mw-redirect" title="Hugh Jones (reverend)">Octave computation</a> (1753) Jones concluded: "Arithmetic by <i>Octaves</i> seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic."<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></li> <li>In 1801, <a href="/wiki/James_Anderson_of_Hermiston" title="James Anderson of Hermiston">James Anderson</a> criticized the French for basing the <a href="/wiki/Metric_system" title="Metric system">metric system</a> on decimal arithmetic. He suggested base 8, for which he coined the term <i>octal</i>. His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of <a href="/wiki/English_units" title="English units">English units</a> was already, to a remarkable extent, an octal system.<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></li> <li>In the mid-19th century, Alfred B. Taylor concluded that "Our octonary [base 8] <a href="/wiki/Radix" title="Radix">radix</a> is, therefore, beyond all comparison the "<i>best possible one</i>" for an arithmetical system." The proposal included a graphical notation for the digits and new names for the numbers, suggesting that we should count "<i>un</i>, <i>du</i>, <i>the</i>, <i>fo</i>, <i>pa</i>, <i>se</i>, <i>ki</i>, <i>unty</i>, <i>unty-un</i>, <i>unty-du</i>" and so on, with successive multiples of eight named "<i>unty</i>, <i>duty</i>, <i>thety</i>, <i>foty</i>, <i>paty</i>, <i>sety</i>, <i>kity</i> and <i>under</i>." So, for example, the number 65 (101 in octal) would be spoken in octonary as <i>under-un</i>.<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><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> Taylor also republished some of Swedenborg's work on octal as an appendix to the above-cited publications.</li></ul> <div class="mw-heading mw-heading3"><h3 id="In_computers"><span class="anchor" id="\nnn"></span>In computers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=5" title="Edit section: In computers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Octal became widely used in computing when systems such as the <a href="/wiki/UNIVAC_1050" title="UNIVAC 1050">UNIVAC 1050</a>, <a href="/wiki/PDP-8" title="PDP-8">PDP-8</a>, <a href="/wiki/ICT_1900_series" title="ICT 1900 series">ICL 1900</a> and <a href="/wiki/IBM_mainframe" title="IBM mainframe">IBM mainframes</a> employed <a href="/wiki/Six-bit_character_code" title="Six-bit character code">6-bit</a>, <a href="/wiki/12-bit_computing" title="12-bit computing">12-bit</a>, <a href="/wiki/24-bit_computing" title="24-bit computing">24-bit</a> or <a href="/wiki/36-bit_computing" title="36-bit computing">36-bit</a> words. Octal was an ideal abbreviation of binary for these machines because their word size is divisible by three (each octal digit represents three binary digits). So two, four, eight or twelve digits could concisely display an entire <a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">machine word</a>. It also cut costs by allowing <a href="/wiki/Nixie_tube" title="Nixie tube">Nixie tubes</a>, <a href="/wiki/Seven-segment_display" title="Seven-segment display">seven-segment displays</a>, and <a href="/wiki/Calculator" title="Calculator">calculators</a> to be used for the operator consoles, where binary displays were too complex to use, decimal displays needed complex hardware to convert radices, and <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> displays needed to display more numerals. </p><p>All modern computing platforms, however, use 16-, 32-, or 64-bit words, further divided into <a href="/wiki/Octet_(computing)" title="Octet (computing)">eight-bit bytes</a>. On such systems three octal digits per byte would be required, with the most significant octal digit representing two binary digits (plus one bit of the next significant byte, if any). Octal representation of a 16-bit word requires 6 digits, but the most significant octal digit represents (quite inelegantly) only one bit (0 or 1). This representation offers no way to easily read the most significant byte, because it's smeared over four octal digits. Therefore, hexadecimal is more commonly used in programming languages today, since two hexadecimal digits exactly specify one byte. Some platforms with a power-of-two word size still have instruction subwords that are more easily understood if displayed in octal; this includes the <a href="/wiki/PDP-11" title="PDP-11">PDP-11</a> and <a href="/wiki/Motorola_68000_family" class="mw-redirect" title="Motorola 68000 family">Motorola 68000 family</a>. The modern-day ubiquitous <a href="/wiki/X86_architecture" class="mw-redirect" title="X86 architecture">x86 architecture</a> belongs to this category as well, but octal is rarely used on this platform, although certain properties of the binary encoding of opcodes become more readily apparent when displayed in octal, e.g. the ModRM byte, which is divided into fields of 2, 3, and 3 bits, so octal can be useful in describing these encodings. Before the availability of <a href="/wiki/Assembler_(computing)" class="mw-redirect" title="Assembler (computing)">assemblers</a>, some programmers would handcode programs in octal; for instance, Dick Whipple and John Arnold wrote <a href="/wiki/Tiny_BASIC" title="Tiny BASIC">Tiny BASIC Extended</a> directly in machine code, using octal.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Octal is sometimes used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with <a href="/wiki/File_permissions" class="mw-redirect" title="File permissions">file permissions</a> under <a href="/wiki/Unix" title="Unix">Unix</a> systems (see <a href="/wiki/Chmod" title="Chmod">chmod</a>). It has the advantage of not requiring any extra symbols as digits (the hexadecimal system is base-16 and therefore needs six additional symbols beyond 0–9). </p><p>In programming languages, octal <a href="/wiki/Literal_(computer_programming)" title="Literal (computer programming)">literals</a> are typically identified with a variety of <a href="/wiki/Prefix" title="Prefix">prefixes</a>, including the digit <code>0</code>, the letters <code>o</code> or <code>q</code>, the digit–letter combination <code>0o</code>, or the symbol <code>&</code><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> or <code>$</code>. In <i>Motorola convention</i>, octal numbers are prefixed with <code>@</code>, whereas a small (or capital<sup id="cite_ref-DRI_1983_CPM86-PG_13-0" class="reference"><a href="#cite_note-DRI_1983_CPM86-PG-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>) letter <code>o</code><sup id="cite_ref-DRI_1983_CPM86-PG_13-1" class="reference"><a href="#cite_note-DRI_1983_CPM86-PG-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> or <code>q</code><sup id="cite_ref-DRI_1983_CPM86-PG_13-2" class="reference"><a href="#cite_note-DRI_1983_CPM86-PG-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> is added as a <a href="/wiki/Suffix" title="Suffix">postfix</a> following the <i>Intel convention</i>.<sup id="cite_ref-Kueveler-Schwoch_1996_14-0" class="reference"><a href="#cite_note-Kueveler-Schwoch_1996-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kueveler-Schwoch_2007_15-0" class="reference"><a href="#cite_note-Kueveler-Schwoch_2007-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Concurrent_DOS" class="mw-redirect" title="Concurrent DOS">Concurrent DOS</a>, <a href="/wiki/Multiuser_DOS" title="Multiuser DOS">Multiuser DOS</a> and <a href="/wiki/REAL/32" class="mw-redirect" title="REAL/32">REAL/32</a> as well as in <a href="/wiki/DOS_Plus" title="DOS Plus">DOS Plus</a> and <a href="/wiki/DR-DOS" title="DR-DOS">DR-DOS</a> various <a href="/wiki/Environment_variable" title="Environment variable">environment variables</a> like <a href="/wiki/%25$CLS%25" class="mw-redirect" title="%$CLS%">$CLS</a>, <a href="/wiki/%25$ON%25" class="mw-redirect" title="%$ON%">$ON</a>, <a href="/wiki/%25$OFF%25" class="mw-redirect" title="%$OFF%">$OFF</a>, <a href="/wiki/%25$HEADER%25" class="mw-redirect" title="%$HEADER%">$HEADER</a> or <a href="/wiki/%25$FOOTER%25" class="mw-redirect" title="%$FOOTER%">$FOOTER</a> support an <code>\nnn</code> octal number notation,<sup id="cite_ref-Paul_1997_NWDOSTIP_16-0" class="reference"><a href="#cite_note-Paul_1997_NWDOSTIP-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Paul_2002_CLS_17-0" class="reference"><a href="#cite_note-Paul_2002_CLS-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CCI_1997_HELP_18-0" class="reference"><a href="#cite_note-CCI_1997_HELP-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and DR-DOS <a href="/wiki/DEBUG" class="mw-redirect" title="DEBUG">DEBUG</a> utilizes <code>\</code> to prefix octal numbers as well. </p><p>For example, the literal 73 (base 8) might be represented as <code>073</code>, <code>o73</code>, <code>q73</code>, <code>0o73</code>, <code>\73</code>, <code>@73</code>, <code>&73</code>, <code>$73</code> or <code>73o</code> in various languages. </p><p>Newer languages have been abandoning the prefix <code>0</code>, as decimal numbers are often represented with leading zeroes. The prefix <code>q</code> was introduced to avoid the prefix <code>o</code> being mistaken for a zero, while the prefix <code>0o</code> was introduced to avoid starting a numerical literal with an alphabetic character (like <code>o</code> or <code>q</code>), since these might cause the literal to be confused with a variable name. The prefix <code>0o</code> also follows the model set by the prefix <code>0x</code> used for hexadecimal literals in the <a href="/wiki/C_(programming_language)" title="C (programming language)">C language</a>; it is supported by <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/OCaml" title="OCaml">OCaml</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> as of version 3.0,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Raku_(programming_language)" title="Raku (programming language)">Raku</a>,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Tcl" title="Tcl">Tcl</a> as of version 9,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <a href="/wiki/PHP" title="PHP">PHP</a> as of version 8.1,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/ECMAScript" title="ECMAScript">ECMAScript</a> as of ECMAScript 6<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> (the prefix <code>0</code> originally stood for base 8 in <a href="/wiki/JavaScript" title="JavaScript">JavaScript</a> but could cause confusion,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> therefore it has been discouraged in ECMAScript 3 and dropped in ECMAScript 5<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>). </p><p>Octal numbers that are used in some programming languages (C, <a href="/wiki/Perl" title="Perl">Perl</a>, <a href="/wiki/PostScript" title="PostScript">PostScript</a>...) for textual/graphical representations of byte strings when some byte values (unrepresented in a code page, non-graphical, having special meaning in current context or otherwise undesired) have to be to <a href="/wiki/Escape_character" title="Escape character">escaped</a> as <code>\nnn</code>. Octal representation may be particularly handy with non-ASCII bytes of <a href="/wiki/UTF-8" title="UTF-8">UTF-8</a>, which encodes groups of 6 bits, and where any start byte has octal value <code>\3nn</code> and any continuation byte has octal value <code>\2nn</code>. </p><p>Octal was also used for <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating point</a> in the <a href="/wiki/Ferranti_Atlas" class="mw-redirect" title="Ferranti Atlas">Ferranti Atlas</a> (1962), <a href="/wiki/Burroughs_B5500" class="mw-redirect" title="Burroughs B5500">Burroughs B5500</a> (1964), <a href="/wiki/Burroughs_B5700" class="mw-redirect" title="Burroughs B5700">Burroughs B5700</a> (1971), <a href="/wiki/Burroughs_B6700" class="mw-redirect" title="Burroughs B6700">Burroughs B6700</a> (1971) and <a href="/wiki/Burroughs_B7700" class="mw-redirect" title="Burroughs B7700">Burroughs B7700</a> (1972) computers. </p> <div class="mw-heading mw-heading3"><h3 id="In_aviation">In aviation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=6" title="Edit section: In aviation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Transponder_(aeronautics)" title="Transponder (aeronautics)">Transponders</a> in aircraft transmit a "squawk" <a href="/wiki/Transponder_(aeronautics)#Transponder_codes" title="Transponder (aeronautics)">code</a>, expressed as a four-octal-digit number, when interrogated by ground radar. This code is used to distinguish different aircraft on the radar screen. </p> <div class="mw-heading mw-heading2"><h2 id="Conversion_between_bases">Conversion between bases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=7" title="Edit section: Conversion between bases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Decimal_to_octal_conversion">Decimal to octal conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=8" title="Edit section: Decimal to octal conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Method_of_successive_Euclidean_division_by_8">Method of successive Euclidean division by 8</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=9" title="Edit section: Method of successive Euclidean division by 8"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert integer decimals to octal, <a href="/wiki/Euclidean_division" title="Euclidean division">divide</a> the original number by the largest possible power of 8 and divide the remainders by successively smaller powers of 8 until the power is 1. The octal representation is formed by the quotients, written in the order generated by the algorithm. For example, to convert 125<sub>10</sub> to octal: </p> <dl><dd>125 = 8<sup>2</sup> × <b>1</b> + 61</dd> <dd>61 = 8<sup>1</sup> × <b>7</b> + 5</dd> <dd>5 = 8<sup>0</sup> × <b>5</b> + 0</dd></dl> <p>Therefore, 125<sub>10</sub> = 175<sub>8</sub>. </p><p>Another example: </p> <dl><dd>900 = 8<sup>3</sup> × <b>1</b> + 388</dd> <dd>388 = 8<sup>2</sup> × <b>6</b> + 4</dd> <dd>4 = 8<sup>1</sup> × <b>0</b> + 4</dd> <dd>4 = 8<sup>0</sup> × <b>4</b> + 0</dd></dl> <p>Therefore, 900<sub>10</sub> = 1604<sub>8</sub>. </p> <div class="mw-heading mw-heading4"><h4 id="Method_of_successive_multiplication_by_8">Method of successive multiplication by 8</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=10" title="Edit section: Method of successive multiplication by 8"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds. </p><p>Example: Convert 0.1640625 to octal: </p> <dl><dd>0.1640625 × 8 = 1.3125 = <b>1</b> + 0.3125</dd> <dd>0.3125 × 8 = 2.5 = <b>2</b> + 0.5</dd> <dd>0.5 × 8 = 4.0 = <b>4</b> + 0</dd></dl> <p>Therefore, 0.1640625<sub>10</sub> = 0.124<sub>8</sub>. </p><p>These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part. </p> <div class="mw-heading mw-heading4"><h4 id="Method_of_successive_duplication">Method of successive duplication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=11" title="Edit section: Method of successive duplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert integer decimals to octal, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix: Double the value to the left side of the radix, using <i>octal</i> rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. If the moved radix point crosses over a digit that is 8 or 9, convert it to 0 or 1 and add the carry to the next leftward digit of the current value. <i>Add</i> <i>octally</i> those digits to the left of the radix and simply drop down those digits to the right, without modification. </p><p>Example: </p> <pre> 0.4 9 1 8 decimal value +0 --------- 4.9 1 8 +1 0 -------- 6 1.1 8 +1 4 2 -------- 7 5 3.8 +1 7 2 6 -------- 1 1 4 6 6. octal value </pre> <div class="mw-heading mw-heading3"><h3 id="Octal_to_decimal_conversion">Octal to decimal conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=12" title="Edit section: Octal to decimal conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert a number <span class="texhtml mvar" style="font-style:italic;">k</span> to decimal, use the formula that defines its base-8 representation: </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 k=\sum _{i=0}^{n}\left(a_{i}\times 8^{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msup> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\sum _{i=0}^{n}\left(a_{i}\times 8^{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d9ec7cb0c2ca3fbe4fb30be97b473232a04136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.014ex; height:6.843ex;" alt="{\displaystyle k=\sum _{i=0}^{n}\left(a_{i}\times 8^{i}\right)}" /></span></dd></dl> <p>In this formula, <span class="texhtml"><i>a</i><sub><i>i</i></sub></span> is an individual octal digit being converted, where <span class="texhtml mvar" style="font-style:italic;">i</span> is the position of the digit (counting from 0 for the right-most digit). </p><p>Example: Convert 764<sub>8</sub> to decimal: </p> <dl><dd>764<sub>8</sub> = 7 × 8<sup>2</sup> + 6 × 8<sup>1</sup> + 4 × 8<sup>0</sup> = 448 + 48 + 4 = 500<sub>10</sub></dd></dl> <p>For double-digit octal numbers this method amounts to multiplying the lead digit by 8 and adding the second digit to get the total. </p><p>Example: 65<sub>8</sub> = 6 × 8 + 5 = 53<sub>10</sub> </p> <div class="mw-heading mw-heading4"><h4 id="Method_of_successive_duplication_2">Method of successive duplication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=13" title="Edit section: Method of successive duplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert octals to decimals, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix: Double the value to the left side of the radix, using <i>decimal</i> rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. <i>Subtract</i> <i>decimally</i> those digits to the left of the radix and simply drop down those digits to the right, without modification. </p><p>Example: </p> <pre> 0.1 1 4 6 6 octal value -0 ----------- 1.1 4 6 6 - 2 ---------- 9.4 6 6 - 1 8 ---------- 7 6.6 6 - 1 5 2 ---------- 6 1 4.6 - 1 2 2 8 ---------- 4 9 1 8. decimal value </pre> <div class="mw-heading mw-heading3"><h3 id="Octal_to_binary_conversion">Octal to binary conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=14" title="Edit section: Octal to binary conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To convert octal to binary, replace each octal digit by its binary representation. </p><p>Example: Convert 51<sub>8</sub> to binary: </p> <dl><dd>5<sub>8</sub> = 101<sub>2</sub></dd> <dd>1<sub>8</sub> = 001<sub>2</sub></dd></dl> <p>Therefore, 51<sub>8</sub> = 101 001<sub>2</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Binary_to_octal_conversion">Binary to octal conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=15" title="Edit section: Binary to octal conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The process is the reverse of the previous algorithm. The binary digits are grouped by threes, starting from the least significant bit and proceeding to the left and to the right. Add leading zeroes (or trailing zeroes to the right of decimal point) to fill out the last group of three if necessary. Then replace each trio with the equivalent octal digit. </p><p>For instance, convert binary 1010111100 to octal: </p> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>001</td> <td>010</td> <td>111</td> <td>100 </td></tr> <tr align="center"> <td>1</td> <td>2</td> <td>7</td> <td>4 </td></tr></tbody></table></dd></dl> <p>Therefore, 1010111100<sub>2</sub> = 1274<sub>8</sub>. </p><p>Convert binary 11100.01001 to octal: </p> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>011</td> <td>100</td> <td> . </td> <td>010</td> <td>010 </td></tr> <tr align="center"> <td>3</td> <td>4</td> <td> . </td> <td>2</td> <td>2 </td></tr></tbody></table></dd></dl> <p>Therefore, 11100.01001<sub>2</sub> = 34.22<sub>8</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Octal_to_hexadecimal_conversion">Octal to hexadecimal conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=16" title="Edit section: Octal to hexadecimal conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conversion is made in two steps using binary as an intermediate base. Octal is converted to binary and then binary to hexadecimal, grouping digits by fours, which correspond each to a hexadecimal digit. </p><p>For instance, convert octal 1057 to hexadecimal: </p> <dl><dd>To binary:</dd></dl> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>1</td> <td>0</td> <td>5</td> <td>7 </td></tr> <tr align="center"> <td>001</td> <td>000</td> <td>101</td> <td>111 </td></tr></tbody></table></dd></dl> <dl><dd>then to hexadecimal:</dd></dl> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>0010</td> <td>0010</td> <td>1111 </td></tr> <tr align="center"> <td>2</td> <td>2</td> <td>F </td></tr></tbody></table></dd></dl> <p>Therefore, 1057<sub>8</sub> = 22F<sub>16</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Hexadecimal_to_octal_conversion">Hexadecimal to octal conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=17" title="Edit section: Hexadecimal to octal conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hexadecimal to octal conversion proceeds by first converting the hexadecimal digits to 4-bit binary values, then regrouping the binary bits into 3-bit octal digits. </p><p>For example, to convert 3FA5<sub>16</sub>: </p> <dl><dd>To binary:</dd></dl> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>3</td> <td>F</td> <td>A</td> <td>5 </td></tr> <tr align="center"> <td>0011</td> <td>1111</td> <td>1010</td> <td>0101 </td></tr></tbody></table></dd></dl> <dl><dd>then to octal:</dd></dl> <dl><dd><table border="1" cellspacing="0" cellpadding="4"> <tbody><tr align="center"> <td>0</td> <td>011</td> <td>111</td> <td>110</td> <td>100</td> <td>101 </td></tr> <tr align="center"> <td>0</td> <td>3</td> <td>7</td> <td>6</td> <td>4</td> <td>5 </td></tr></tbody></table></dd></dl> <p>Therefore, 3FA5<sub>16</sub> = 37645<sub>8</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Real_numbers">Real numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=18" title="Edit section: Real numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fractions">Fractions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=19" title="Edit section: Fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Due to having only factors of two, many octal fractions have repeating digits, although these tend to be fairly simple: </p> <table class="wikitable"> <tbody><tr> <td colspan="3" align="center">Decimal base<br /><small>Prime factors of the base: <span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span></small><br /><small>Prime factors of one below the base: <span style="color:Blue"><b>3</b></span></small><br /><small>Prime factors of one above the base: <span style="color:Magenta"><b>11</b></span></small><br /><small>Other Prime factors: <span style="color:Red"><b>7 13 17 19 23 29 31</b></span></small> </td> <td colspan="3" align="center"><b>Octal base</b><br /><small>Prime factors of the base: <span style="color:Green"><b>2</b></span></small><br /><small>Prime factors of one below the base: <span style="color:Blue"><b>7</b></span></small><br /><small>Prime factors of one above the base: <span style="color:Magenta"><b>3</b></span></small><br /><small>Other Prime factors: <span style="color:Red"><b>5 13 15 21 23 27 35 37</b></span></small> </td></tr> <tr> <td align="center">Fraction </td> <td align="center"><small>Prime factors<br />of the denominator</small> </td> <td align="center">Positional representation </td> <td align="center">Positional representation </td> <td align="center"><small>Prime factors<br />of the denominator</small> </td> <td align="center">Fraction </td></tr> <tr> <td align="center">1/2 </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.5</b> </td> <td><b>0.4</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center">1/2 </td></tr> <tr> <td align="center">1/3 </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b>3333... = <b>0.</b><span style="text-decoration:overline;">3</span> </td> <td bgcolor="#c0c0c0"><b>0.</b>2525... = <b>0.</b><span style="text-decoration:overline;">25</span> </td> <td align="center"><span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/3 </td></tr> <tr> <td align="center">1/4 </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.25</b> </td> <td><b>0.2</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center">1/4 </td></tr> <tr> <td align="center">1/5 </td> <td align="center"><span style="color:Green"><b>5</b></span> </td> <td><b>0.2</b> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">1463</span> </td> <td align="center"><span style="color:Red"><b>5</b></span> </td> <td align="center">1/5 </td></tr> <tr> <td align="center">1/6 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.1</b><span style="text-decoration:overline;">6</span> </td> <td bgcolor="#c0c0c0"><b>0.1</b><span style="text-decoration:overline;">25</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/6 </td></tr> <tr> <td align="center">1/7 </td> <td align="center"><span style="color:Red"><b>7</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">142857</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">1</span> </td> <td align="center"><span style="color:Blue"><b>7</b></span> </td> <td align="center">1/7 </td></tr> <tr> <td align="center">1/8 </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.125</b> </td> <td><b>0.1</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center">1/10 </td></tr> <tr> <td align="center">1/9 </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">1</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">07</span> </td> <td align="center"><span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/11 </td></tr> <tr> <td align="center">1/10 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span> </td> <td><b>0.1</b> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">6314</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>5</b></span> </td> <td align="center">1/12 </td></tr> <tr> <td align="center">1/11 </td> <td align="center"><span style="color:Magenta"><b>11</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">09</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0564272135</span> </td> <td align="center"><span style="color:Red"><b>13</b></span> </td> <td align="center">1/13 </td></tr> <tr> <td align="center">1/12 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.08</b><span style="text-decoration:overline;">3</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">52</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/14 </td></tr> <tr> <td align="center">1/13 </td> <td align="center"><span style="color:Red"><b>13</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">076923</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0473</span> </td> <td align="center"><span style="color:Red"><b>15</b></span> </td> <td align="center">1/15 </td></tr> <tr> <td align="center">1/14 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">714285</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">4</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>7</b></span> </td> <td align="center">1/16 </td></tr> <tr> <td align="center">1/15 </td> <td align="center"><span style="color:Blue"><b>3</b></span>, <span style="color:Green"><b>5</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">6</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0421</span> </td> <td align="center"><span style="color:Magenta"><b>3</b></span>, <span style="color:Red"><b>5</b></span> </td> <td align="center">1/17 </td></tr> <tr> <td align="center">1/16 </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.0625</b> </td> <td><b>0.04</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center">1/20 </td></tr> <tr> <td align="center">1/17 </td> <td align="center"><span style="color:Red"><b>17</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0588235294117647</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">03607417</span> </td> <td align="center"><span style="color:Red"><b>21</b></span> </td> <td align="center">1/21 </td></tr> <tr> <td align="center">1/18 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">5</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">34</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/22 </td></tr> <tr> <td align="center">1/19 </td> <td align="center"><span style="color:Red"><b>19</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">052631578947368421</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">032745</span> </td> <td align="center"><span style="color:Red"><b>23</b></span> </td> <td align="center">1/23 </td></tr> <tr> <td align="center">1/20 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span> </td> <td><b>0.05</b> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">3146</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>5</b></span> </td> <td align="center">1/24 </td></tr> <tr> <td align="center">1/21 </td> <td align="center"><span style="color:Blue"><b>3</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">047619</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">03</span> </td> <td align="center"><span style="color:Magenta"><b>3</b></span>, <span style="color:Blue"><b>7</b></span> </td> <td align="center">1/25 </td></tr> <tr> <td align="center">1/22 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">45</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">2721350564</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>13</b></span> </td> <td align="center">1/26 </td></tr> <tr> <td align="center">1/23 </td> <td align="center"><span style="color:Red"><b>23</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0434782608695652173913</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">02620544131</span> </td> <td align="center"><span style="color:Red"><b>27</b></span> </td> <td align="center">1/27 </td></tr> <tr> <td align="center">1/24 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.041</b><span style="text-decoration:overline;">6</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">25</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/30 </td></tr> <tr> <td align="center">1/25 </td> <td align="center"><span style="color:Green"><b>5</b></span> </td> <td><b>0.04</b> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">02436560507534121727</span> </td> <td align="center"><span style="color:Red"><b>5</b></span> </td> <td align="center">1/31 </td></tr> <tr> <td align="center">1/26 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>13</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">384615</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">2354</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>15</b></span> </td> <td align="center">1/32 </td></tr> <tr> <td align="center">1/27 </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">037</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">022755</span> </td> <td align="center"><span style="color:Magenta"><b>3</b></span> </td> <td align="center">1/33 </td></tr> <tr> <td align="center">1/28 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#c0c0c0"><b>0.03</b><span style="text-decoration:overline;">571428</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">2</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>7</b></span> </td> <td align="center">1/34 </td></tr> <tr> <td align="center">1/29 </td> <td align="center"><span style="color:Red"><b>29</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0344827586206896551724137931</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">0215173454106475626043236713</span> </td> <td align="center"><span style="color:Red"><b>35</b></span> </td> <td align="center">1/35 </td></tr> <tr> <td align="center">1/30 </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span>, <span style="color:Green"><b>5</b></span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">3</span> </td> <td bgcolor="#c0c0c0"><b>0.0</b><span style="text-decoration:overline;">2104</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>3</b></span>, <span style="color:Red"><b>5</b></span> </td> <td align="center">1/36 </td></tr> <tr> <td align="center">1/31 </td> <td align="center"><span style="color:Red"><b>31</b></span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">032258064516129</span> </td> <td bgcolor="#c0c0c0"><b>0.</b><span style="text-decoration:overline;">02041</span> </td> <td align="center"><span style="color:Red"><b>37</b></span> </td> <td align="center">1/37 </td></tr> <tr> <td align="center">1/32 </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.03125</b> </td> <td><b>0.02</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center">1/40 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Irrational_numbers">Irrational numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Octal&action=edit&section=20" title="Edit section: Irrational numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table below gives the expansions of some common <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> in decimal and octal. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Number </th> <th colspan="2">Positional representation </th></tr> <tr> <th>Decimal </th> <th>Octal </th></tr> <tr> <td><a href="/wiki/Square_root_of_2" title="Square root of 2"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></a> <span style="font-size: 85%;">(the length of the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> of a unit <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a>)</span> </td> <td><span class="nowrap"><span data-sort-value="7000141421356237309♠"></span>1.414<span style="margin-left:.25em;">213</span><span style="margin-left:.25em;">562</span><span style="margin-left:.25em;">373</span><span style="margin-left:.25em;">095</span><span style="margin-left:.25em;">048</span></span>... </td> <td>1.3240 4746 3177 1674... </td></tr> <tr> <td><a href="/wiki/Square_root_of_3" title="Square root of 3"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span></a> <span style="font-size: 85%;">(the length of the diagonal of a unit <a href="/wiki/Cube" title="Cube">cube</a>)</span> </td> <td><span class="nowrap"><span data-sort-value="7000173205080756887♠"></span>1.732<span style="margin-left:.25em;">050</span><span style="margin-left:.25em;">807</span><span style="margin-left:.25em;">568</span><span style="margin-left:.25em;">877</span><span style="margin-left:.25em;">293</span></span>... </td> <td>1.5666 3656 4130 2312... </td></tr> <tr> <td><a href="/wiki/Square_root_of_5" title="Square root of 5"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span></a> <span style="font-size: 85%;">(the length of the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> of a 1×2 <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>)</span> </td> <td><span class="nowrap"><span data-sort-value="7000223606797749978♠"></span>2.236<span style="margin-left:.25em;">067</span><span style="margin-left:.25em;">977</span><span style="margin-left:.25em;">499</span><span style="margin-left:.25em;">789</span><span style="margin-left:.25em;">696</span></span>... </td> <td>2.1706 7363 3457 7224... </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Golden_ratio" title="Golden ratio">φ</a></span> <span style="font-size: 85%;">(phi, the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> = <span class="texhtml">(1+<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>)/2</span>)</span> </td> <td><span class="nowrap"><span data-sort-value="7000161803398874989♠"></span>1.618<span style="margin-left:.25em;">033</span><span style="margin-left:.25em;">988</span><span style="margin-left:.25em;">749</span><span style="margin-left:.25em;">894</span><span style="margin-left:.25em;">848</span></span>... </td> <td>1.4743 3571 5627 7512... </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Pi" title="Pi">π</a></span> <span style="font-size: 85%;">(pi, the ratio of <a href="/wiki/Circumference" title="Circumference">circumference</a> to <a href="/wiki/Diameter" title="Diameter">diameter</a> of a circle)</span> </td> <td><span class="nowrap"><span data-sort-value="7000314159265358979♠"></span>3.141<span style="margin-left:.25em;">592</span><span style="margin-left:.25em;">653</span><span style="margin-left:.25em;">589</span><span style="margin-left:.25em;">793</span><span style="margin-left:.25em;">238</span><span style="margin-left:.25em;">462</span><span style="margin-left:.25em;">643</span></span><br /><span class="nowrap"><span data-sort-value="7026383279502884197♠"></span>383<span style="margin-left:.25em;">279</span><span style="margin-left:.25em;">502</span><span style="margin-left:.25em;">884</span><span style="margin-left:.25em;">197</span><span style="margin-left:.25em;">169</span><span style="margin-left:.25em;">399</span><span style="margin-left:.25em;">375</span><span style="margin-left:.25em;">105</span></span>... </td> <td>3.1103 7552 4210 2643... </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span> <span style="font-size: 85%;">(the base of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>)</span> </td> <td><span class="nowrap"><span data-sort-value="7000271828182845904♠"></span>2.718<span style="margin-left:.25em;">281</span><span style="margin-left:.25em;">828</span><span style="margin-left:.25em;">459</span><span style="margin-left:.25em;">045</span><span style="margin-left:.25em;">235</span></span>... </td> <td>2.5576 0521 3050 5355... </td></tr></tbody></table> <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=Octal&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Computer_number_format" title="Computer number format">Computer number format</a> – Internal representation of numeric values in a digital computer</li> <li><a href="/wiki/Octal_games" class="mw-redirect" title="Octal games">Octal games</a>, a game numbering system used in <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a></li> <li><a href="/wiki/Split_octal" title="Split octal">Split octal</a>, a 16-bit octal notation used by the Heath Company, DEC and others</li> <li><a href="/wiki/Squawk_code" class="mw-redirect" title="Squawk code">Squawk code</a>, a 12-bit octal representation of <a href="/wiki/Gillham_code" title="Gillham code">Gillham code</a></li> <li><a href="/wiki/Syllabic_octal" class="mw-redirect" title="Syllabic octal">Syllabic octal</a>, an octal representation of 8-bit syllables used by English Electric</li></ul> <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=Octal&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns 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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="CITEREFLeibniz1703" class="citation web cs1">Leibniz, Gottfried Wilhelm (1703). <a rel="nofollow" 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Retrieved <span class="nowrap">2022-03-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=leibniz-translations.com&rft.atitle=Explanation+of+binary+arithmetic&rft.date=1703&rft.aulast=Leibniz&rft.aufirst=Gottfried+Wilhelm&rft_id=http%3A%2F%2Fwww.leibniz-translations.com%2Fbinary.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAscher1992" class="citation journal cs1"><a href="/wiki/Marcia_Ascher" title="Marcia Ascher">Ascher, Marcia</a> (1992). 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Retrieved <span class="nowrap">2013-06-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Some+thoughts+about+Indo-European+numerals&rft.btitle=Indo-European+numerals&rft.place=Berlin&rft.series=Trends+in+Linguistics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E13-%3C%2Fspan%3E14&rft.pub=Mouton+de+Gruyter&rft.date=1991&rft.isbn=3-11-011322-8&rft.aulast=Winter&rft.aufirst=Werner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DS-hmNOLuDGsC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilkins1668" class="citation book cs1">Wilkins, John (1668). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BCCtZjBtiEYC&pg=PA190"><i>An Essay Towards a Real Character and a Philosophical Language</i></a>. London. p. 190. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401131651/https://books.google.com/books?id=BCCtZjBtiEYC&pg=PA190">Archived</a> from the original on 2023-04-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-02-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Essay+Towards+a+Real+Character+and+a+Philosophical+Language&rft.place=London&rft.pages=190&rft.date=1668&rft.aulast=Wilkins&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBCCtZjBtiEYC%26pg%3DPA190&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>, <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i></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">See H. R. Phalen, "Hugh Jones and Octave Computation," <i>The American Mathematical Monthly</i> 56 (August–September 1949): 461-465.</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">James Anderson, On Octal Arithmetic [title appears only in page headers], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=olhHAAAAYAAJ&pg=PA437">Recreations in Agriculture, Natural-History, Arts, and Miscellaneous Literature</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401131644/https://books.google.com/books?id=olhHAAAAYAAJ&pg=PA437">Archived</a> 2023-04-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Vol. IV, No. 6 (February 1801), T. Bensley, London; pages 437-448.</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">Alfred B. Taylor, <a rel="nofollow" class="external text" href="https://archive.org/details/reportonweights00taylgoog">Report on Weights and Measures</a>, Pharmaceutical Association, 8th Annual Session, Boston, 1859-09-15. See pages 48 and 53.</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">Alfred B. Taylor, Octonary numeration and its application to a system of weights and measures, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296">Proc. Amer. Phil. Soc. Vol XXIV</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401131638/https://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296">Archived</a> 2023-04-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Philadelphia, 1887; pages 296-366. See pages 327 and 330.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation journal cs1">"TB Code Sheet". <i><a href="/wiki/Dr._Dobb%27s_Journal" title="Dr. Dobb's Journal">Dr. Dobb's Journal of Computer Calisthenics & Orthodontia, Running Light Without Overbyte</a></i>. <b>1</b> (1). December 1975.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dr.+Dobb%27s+Journal+of+Computer+Calisthenics+%26+Orthodontia%2C+Running+Light+Without+Overbyte&rft.atitle=TB+Code+Sheet&rft.volume=1&rft.issue=1&rft.date=1975-12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMicrosoft_Corporation1987" class="citation book cs1">Microsoft Corporation (1987). <a rel="nofollow" class="external text" href="http://www.antonis.de/qbebooks/gwbasman/chapter%206.html">"Constants, Variables, Expressions and Operators"</a>. <a rel="nofollow" class="external text" href="http://www.antonis.de/qbebooks/gwbasman/"><i>GW-BASIC User's Manual</i></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160105062314/http://www.antonis.de/qbebooks/gwbasman/">Archived</a> from the original on 2016-01-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-12-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Constants%2C+Variables%2C+Expressions+and+Operators&rft.btitle=GW-BASIC+User%27s+Manual&rft.date=1987&rft.au=Microsoft+Corporation&rft_id=http%3A%2F%2Fwww.antonis.de%2Fqbebooks%2Fgwbasman%2Fchapter%25206.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-DRI_1983_CPM86-PG-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-DRI_1983_CPM86-PG_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DRI_1983_CPM86-PG_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-DRI_1983_CPM86-PG_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">"2.4.1 Numeric Constants". <a rel="nofollow" class="external text" href="http://www.bitsavers.org/pdf/digitalResearch/cpm-86/CPM-86_Programmers_Guide_Jan83.pdf"><i>CP/M-86 - Operating System - Programmer's Guide</i></a> <span class="cs1-format">(PDF)</span> (3 ed.). Pacific Grove, California, USA: <a href="/wiki/Digital_Research" title="Digital Research">Digital Research</a>. January 1983 [1981]. p. 9. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200227225328/http://www.bitsavers.org/pdf/digitalResearch/cpm-86/CPM-86_Programmers_Guide_Jan83.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2020-02-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-02-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.4.1+Numeric+Constants&rft.btitle=CP%2FM-86+-+Operating+System+-+Programmer%27s+Guide&rft.place=Pacific+Grove%2C+California%2C+USA&rft.pages=9&rft.edition=3&rft.pub=Digital+Research&rft.date=1983-01&rft_id=http%3A%2F%2Fwww.bitsavers.org%2Fpdf%2FdigitalResearch%2Fcpm-86%2FCPM-86_Programmers_Guide_Jan83.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span> <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20200227132338/https://www.z80cpu.eu/mirrors/oldcomputers.dyndns.org/public/pub/manuals/cpm86pg.pdf">[1]</a> (1+viii+122+2 pages)</span> </li> <li id="cite_note-Kueveler-Schwoch_1996-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kueveler-Schwoch_1996_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKüvelerSchwoch2013" class="citation book cs1 cs1-prop-foreign-lang-source">Küveler, Gerd; Schwoch, Dietrich (2013) [1996]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b8-dBgAAQBAJ"><i>Arbeitsbuch Informatik - eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe</i></a> (in German). Vieweg-Verlag, reprint: Springer-Verlag. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-322-92907-5">10.1007/978-3-322-92907-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-528-04952-2" title="Special:BookSources/978-3-528-04952-2"><bdi>978-3-528-04952-2</bdi></a>. 978-3-32292907-5. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401131645/https://books.google.com/books?id=b8-dBgAAQBAJ">Archived</a> from the original on 2023-04-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arbeitsbuch+Informatik+-+eine+praxisorientierte+Einf%C3%BChrung+in+die+Datenverarbeitung+mit+Projektaufgabe&rft.pub=Vieweg-Verlag%2C+reprint%3A+Springer-Verlag&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-3-322-92907-5&rft.isbn=978-3-528-04952-2&rft.aulast=K%C3%BCveler&rft.aufirst=Gerd&rft.au=Schwoch%2C+Dietrich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Db8-dBgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-Kueveler-Schwoch_2007-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kueveler-Schwoch_2007_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKüvelerSchwoch2007" class="citation book cs1 cs1-prop-foreign-lang-source">Küveler, Gerd; Schwoch, Dietrich (2007-10-04). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xQbvPYxceY0C"><i>Informatik für Ingenieure und Naturwissenschaftler: PC- und Mikrocomputertechnik, Rechnernetze</i></a> (in German). Vol. 2 (5 ed.). Vieweg, reprint: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-83489191-4" title="Special:BookSources/978-3-83489191-4"><bdi>978-3-83489191-4</bdi></a>. 978-3-83489191-4. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230401131639/https://books.google.com/books?id=xQbvPYxceY0C">Archived</a> from the original on 2023-04-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Informatik+f%C3%BCr+Ingenieure+und+Naturwissenschaftler%3A+PC-+und+Mikrocomputertechnik%2C+Rechnernetze&rft.edition=5&rft.pub=Vieweg%2C+reprint%3A+Springer-Verlag&rft.date=2007-10-04&rft.isbn=978-3-83489191-4&rft.aulast=K%C3%BCveler&rft.aufirst=Gerd&rft.au=Schwoch%2C+Dietrich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxQbvPYxceY0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-Paul_1997_NWDOSTIP-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Paul_1997_NWDOSTIP_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPaul1997" class="citation book cs1 cs1-prop-foreign-lang-source">Paul, Matthias R. (1997-07-30). <a rel="nofollow" class="external text" href="http://www.antonis.de/dos/dos-tuts/mpdostip/html/nwdostip.htm">"NWDOS-TIPs — Tips & Tricks rund um Novell DOS 7, mit Blick auf undokumentierte Details, Bugs und Workarounds"</a>. <i>MPDOSTIP</i>. Release 157 (in German) (3 ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161104235829/http://www.antonis.de/dos/dos-tuts/mpdostip/html/nwdostip.htm">Archived</a> from the original on 2016-11-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=NWDOS-TIPs+%E2%80%94+Tips+%26+Tricks+rund+um+Novell+DOS+7%2C+mit+Blick+auf+undokumentierte+Details%2C+Bugs+und+Workarounds&rft.btitle=MPDOSTIP&rft.series=Release+157&rft.edition=3&rft.date=1997-07-30&rft.aulast=Paul&rft.aufirst=Matthias+R.&rft_id=http%3A%2F%2Fwww.antonis.de%2Fdos%2Fdos-tuts%2Fmpdostip%2Fhtml%2Fnwdostip.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span> (NB. NWDOSTIP.TXT is a comprehensive work on <a href="/wiki/Novell_DOS_7" class="mw-redirect" title="Novell DOS 7">Novell DOS 7</a> and <a href="/wiki/OpenDOS_7.01" class="mw-redirect" title="OpenDOS 7.01">OpenDOS 7.01</a>, including the description of many undocumented features and internals. It is part of the author's yet larger <code>MPDOSTIP.ZIP</code> collection maintained up to 2001 and distributed on many sites at the time. The provided link points to a HTML-converted older version of the <code>NWDOSTIP.TXT</code> file.)</span> </li> <li id="cite_note-Paul_2002_CLS-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Paul_2002_CLS_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPaul2002" class="citation web cs1">Paul, Matthias R. (2002-03-26). <a rel="nofollow" class="external text" href="https://marc.info/?l=freedos-dev&m=101717593306186&w=2">"Updated CLS posted"</a>. freedos-dev mailing list. <a rel="nofollow" class="external text" href="https://archive.today/20190427173821/https://marc.info/?l=freedos-dev&m=101717593306186&w=2">Archived</a> from the original on 2019-04-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Updated+CLS+posted&rft.pub=freedos-dev+mailing+list&rft.date=2002-03-26&rft.aulast=Paul&rft.aufirst=Matthias+R.&rft_id=http%3A%2F%2Fmarc.info%2F%3Fl%3Dfreedos-dev%26m%3D101717593306186%26w%3D2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-CCI_1997_HELP-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-CCI_1997_HELP_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><i>CCI Multiuser DOS 7.22 GOLD Online Documentation</i>. <a href="/wiki/Concurrent_Controls,_Inc." class="mw-redirect" title="Concurrent Controls, Inc.">Concurrent Controls, Inc.</a> (CCI). 1997-02-10. HELP.HLP.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CCI+Multiuser+DOS+7.22+GOLD+Online+Documentation&rft.pub=Concurrent+Controls%2C+Inc.+%28CCI%29&rft.date=1997-02-10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.haskell.org/onlinereport/lexemes.html#sect2.5">"Haskell 98 Lexical Structure"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210411061515/https://www.haskell.org/onlinereport/lexemes.html#sect2.5">Archived</a> from the original on 2021-04-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Haskell+98+Lexical+Structure&rft_id=https%3A%2F%2Fwww.haskell.org%2Fonlinereport%2Flexemes.html%23sect2.5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">OCaml: <a rel="nofollow" class="external text" href="http://caml.inria.fr/pub/docs/manual-ocaml/lex.html">7.1 Lexical conventions</a> <a rel="nofollow" class="external text" href="https://archive.today/20130701170755/http://caml.inria.fr/pub/docs/manual-ocaml/lex.html">Archived</a> 2013-07-01 at <a href="/wiki/Archive.today" title="Archive.today">archive.today</a></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Python 3: <a rel="nofollow" class="external free" href="https://docs.python.org/3.1/reference/lexical_analysis.html#integer-literals">https://docs.python.org/3.1/reference/lexical_analysis.html#integer-literals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140320021754/http://docs.python.org/3.1/reference/lexical_analysis.html#integer-literals">Archived</a> 2014-03-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Perl 6: <a rel="nofollow" class="external free" href="http://perlcabal.org/syn/S02.html#Radix_markers">http://perlcabal.org/syn/S02.html#Radix_markers</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141031161348/http://perlcabal.org/syn/S02.html#Radix_markers">Archived</a> 31 October 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">RubySpec: <a rel="nofollow" class="external free" href="https://github.com/ruby/ruby/blob/master/spec/ruby/core/string/to_i_spec.rb">https://github.com/ruby/ruby/blob/master/spec/ruby/core/string/to_i_spec.rb</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220529183751/https://github.com/ruby/ruby/blob/master/spec/ruby/core/string/to_i_spec.rb">Archived</a> 2022-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Tcl: <a rel="nofollow" class="external free" href="http://wiki.tcl.tk/498">http://wiki.tcl.tk/498</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140104045014/http://wiki.tcl.tk/498">Archived</a> 2014-01-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">PHP.Watch - PHP 8.1: Explicit Octal numeral notation <a rel="nofollow" class="external free" href="https://php.watch/versions/8.1/explicit-octal-notation">https://php.watch/versions/8.1/explicit-octal-notation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210108032852/https://php.watch/versions/8.1/explicit-octal-notation">Archived</a> 2021-01-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Rust literals and operators: <a rel="nofollow" class="external free" href="https://doc.rust-lang.org/rust-by-example/primitives/literals.html">https://doc.rust-lang.org/rust-by-example/primitives/literals.html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220528081546/https://doc.rust-lang.org/rust-by-example/primitives/literals.html">Archived</a> 2022-05-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">ECMAScript 6th Edition draft: <a rel="nofollow" class="external free" href="https://people.mozilla.org/~jorendorff/es6-draft.html#sec-literals-numeric-literals">https://people.mozilla.org/~jorendorff/es6-draft.html#sec-literals-numeric-literals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131216202526/https://people.mozilla.org/~jorendorff/es6-draft.html#sec-literals-numeric-literals">Archived</a> 16 December 2013 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stackoverflow.com/questions/5600366/why-does-the-radix-for-javascripts-parseint-default-to-8">"Why does the radix for JavaScript's parseInt default to 8?"</a>. <i>Stack Overflow</i>. 2011-04-08. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200806000322/https://stackoverflow.com/questions/5600366/why-does-the-radix-for-javascripts-parseint-default-to-8">Archived</a> from the original on 2020-08-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Overflow&rft.atitle=Why+does+the+radix+for+JavaScript%27s+parseInt+default+to+8%3F&rft.date=2011-04-08&rft_id=https%3A%2F%2Fstackoverflow.com%2Fquestions%2F5600366%2Fwhy-does-the-radix-for-javascripts-parseint-default-to-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/parseInt">"parseInt()"</a>, <i>Mozilla Developer Network (MDN)</i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140305220352/https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/parseInt">archived</a> from the original on 2014-03-05<span class="reference-accessdate">, retrieved <span class="nowrap">2014-01-03</span></span>, <q>If the input string begins with "0" (a zero), radix is assumed to be 8 (octal) or 10 (decimal). Exactly which radix is chosen is implementation-dependent. ECMAScript 5 clarifies that 10 (decimal) should be used, but not all browsers support this yet</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mozilla+Developer+Network+%28MDN%29&rft.atitle=parseInt%28%29&rft_id=https%3A%2F%2Fdeveloper.mozilla.org%2Fen-US%2Fdocs%2FWeb%2FJavaScript%2FReference%2FGlobal_Objects%2FparseInt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOctal" class="Z3988"></span></span> </li> </ol></div></div> <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=Octal&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.octomatics.org">Octomatics</a> is a <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> enabling simple visual calculation in octal.</li> <li><a rel="nofollow" class="external text" href="https://converter.app/octal-to-decimal/">Octal converter</a> performs bidirectional conversions between the octal and decimal system.</li></ul>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Octal&oldid=1282600110">https://en.wikipedia.org/w/index.php?title=Octal&oldid=1282600110</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Binary_arithmetic" title="Category:Binary arithmetic">Binary arithmetic</a></li><li><a href="/wiki/Category:Power-of-two_numeral_systems" title="Category:Power-of-two numeral systems">Power-of-two numeral systems</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Webarchive_template_archiveis_links" title="Category:Webarchive template archiveis links">Webarchive template archiveis links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_August_2019" title="Category:Use dmy dates from August 2019">Use dmy dates from August 2019</a></li><li><a href="/wiki/Category:Pages_using_sidebar_with_the_child_parameter" title="Category:Pages using sidebar with the child parameter">Pages using sidebar with the child parameter</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 March 2025, at 12:22<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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