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</a> <ul id="toc-Written_representation-sublist" class="vector-toc-list"> <li id="toc-Distinguishing_from_decimal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Distinguishing_from_decimal"> <div class="vector-toc-text"> 1.1.1 Distinguishing from decimal </div> </a> <ul id="toc-Distinguishing_from_decimal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Syntax_that_is_always_Hex" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Syntax_that_is_always_Hex"> <div class="vector-toc-text"> 1.1.2 Syntax that is always Hex </div> </a> <ul id="toc-Syntax_that_is_always_Hex-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_symbols_for_10–15_and_mostly_different_symbol_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_symbols_for_10–15_and_mostly_different_symbol_sets"> <div class="vector-toc-text"> 1.2 Other symbols for 10–15 and mostly different symbol sets </div> </a> <ul id="toc-Other_symbols_for_10–15_and_mostly_different_symbol_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Verbal_and_digital_representations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Verbal_and_digital_representations"> <div class="vector-toc-text"> 1.3 Verbal and digital representations </div> </a> <ul id="toc-Verbal_and_digital_representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signs"> <div class="vector-toc-text"> 1.4 Signs </div> </a> <ul id="toc-Signs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hexadecimal_exponential_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hexadecimal_exponential_notation"> <div class="vector-toc-text"> 1.5 Hexadecimal exponential notation </div> </a> <ul id="toc-Hexadecimal_exponential_notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conversion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conversion"> <div class="vector-toc-text"> 2 Conversion </div> </a> <button aria-controls="toc-Conversion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Conversion subsection </button> <ul id="toc-Conversion-sublist" class="vector-toc-list"> <li id="toc-Binary_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_conversion"> <div class="vector-toc-text"> 2.1 Binary conversion </div> </a> <ul id="toc-Binary_conversion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_simple_conversions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_simple_conversions"> <div class="vector-toc-text"> 2.2 Other simple conversions </div> </a> <ul id="toc-Other_simple_conversions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division-remainder_in_source_base" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division-remainder_in_source_base"> <div class="vector-toc-text"> 2.3 Division-remainder in source base </div> </a> <ul id="toc-Division-remainder_in_source_base-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conversion_through_addition_and_multiplication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conversion_through_addition_and_multiplication"> <div class="vector-toc-text"> 2.4 Conversion through addition and multiplication </div> </a> <ul id="toc-Conversion_through_addition_and_multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tools_for_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tools_for_conversion"> <div class="vector-toc-text"> 2.5 Tools for conversion </div> </a> <ul id="toc-Tools_for_conversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elementary_arithmetic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementary_arithmetic"> <div class="vector-toc-text"> 3 Elementary arithmetic </div> </a> <ul id="toc-Elementary_arithmetic-sublist" class="vector-toc-list"> </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"> 4 Real numbers </div> </a> <button aria-controls="toc-Real_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Real numbers subsection </button> <ul id="toc-Real_numbers-sublist" class="vector-toc-list"> <li id="toc-Rational_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_numbers"> <div class="vector-toc-text"> 4.1 Rational numbers </div> </a> <ul id="toc-Rational_numbers-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"> 4.2 Irrational numbers </div> </a> <ul id="toc-Irrational_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Powers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Powers"> <div class="vector-toc-text"> 4.3 Powers </div> </a> <ul id="toc-Powers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cultural_history" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cultural_history"> <div class="vector-toc-text"> 5 Cultural history </div> </a> <ul id="toc-Cultural_history-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Base16_(transfer_encoding)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Base16_(transfer_encoding)"> <div class="vector-toc-text"> 6 Base16 (transfer encoding) </div> </a> <ul id="toc-Base16_(transfer_encoding)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </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"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" 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" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Hexadecimal</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 81 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-81" aria-hidden="true" > 81 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Heksadesimale_stelsel" title="Heksadesimale stelsel – Afrikaans" lang="af" hreflang="af" data-title="Heksadesimale stelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%B9%D8%AF_%D8%B3%D8%AA%D8%A9_%D8%B9%D8%B4%D8%B1%D9%8A" title="نظام عد ستة عشري – Arabic" lang="ar" hreflang="ar" data-title="نظام عد ستة عشري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%8F%D5%A1%D5%BD%D5%B6%D5%BE%D5%A5%D6%81%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Տասնվեցային համակարգ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Տասնվեցային համակարգ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Onalt%C4%B1l%C4%B1q_say_sistemi" title="Onaltılıq say sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Onaltılıq say sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B7%E0%A7%8B%E0%A6%A1%E0%A6%BC%E0%A6%B6%E0%A6%BF%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="ষোড়শিক সংখ্যা পদ্ধতি – Bangla" lang="bn" hreflang="bn" data-title="ষোড়শিক সংখ্যা পদ্ধতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Cha%CC%8Dp-la%CC%8Dk-ch%C3%ACn-hoat" title="Cha̍p-la̍k-chìn-hoat – Minnan" lang="nan" hreflang="nan" data-title="Cha̍p-la̍k-chìn-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A8%D0%B0%D1%81%D0%BD%D0%B0%D1%86%D1%86%D0%B0%D1%82%D0%BA%D0%BE%D0%B2%D0%B0%D1%8F_%D1%81%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B7%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D0%BD%D1%8F" title="Шаснаццатковая сістэма злічэння – Belarusian" lang="be" hreflang="be" data-title="Шаснаццатковая сістэма злічэння" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A8%D0%B0%D1%81%D0%BD%D0%B0%D1%86%D1%86%D0%B0%D1%82%D0%BA%D0%BE%D0%B2%D0%B0%D1%8F_%D1%81%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%B7%D1%8C%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D1%8C%D0%BD%D1%8F" title="Шаснаццатковая сыстэма зьлічэньня – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Шаснаццатковая сыстэма зьлічэньня" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A8%D0%B5%D1%81%D1%82%D0%BD%D0%B0%D0%B9%D1%81%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B1%D1%80%D0%BE%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Шестнайсетична бройна система – Bulgarian" lang="bg" hreflang="bg" data-title="Шестнайсетична бройна система" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Heksadecimalni_brojevni_sistem" title="Heksadecimalni brojevni sistem – Bosnian" lang="bs" hreflang="bs" data-title="Heksadecimalni brojevni sistem" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Diazez_c%27hwezekred" title="Diazez c'hwezekred – Breton" lang="br" hreflang="br" data-title="Diazez c'hwezekred" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_hexadecimal" title="Sistema hexadecimal – Catalan" lang="ca" hreflang="ca" data-title="Sistema hexadecimal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C5%A0estn%C3%A1ctkov%C3%A1_soustava" title="Šestnáctková soustava – Czech" lang="cs" hreflang="cs" data-title="Šestnáctková soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Hecsadegol" title="Hecsadegol – Welsh" lang="cy" hreflang="cy" data-title="Hecsadegol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hexadecimale_talsystem" title="Hexadecimale talsystem – Danish" lang="da" hreflang="da" data-title="Hexadecimale talsystem" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hexadezimalsystem" title="Hexadezimalsystem – German" lang="de" hreflang="de" data-title="Hexadezimalsystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kuueteistk%C3%BCmnends%C3%BCsteem" title="Kuueteistkümnendsüsteem – Estonian" lang="et" hreflang="et" data-title="Kuueteistkümnendsüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B5%CE%BA%CE%B1%CE%B5%CE%BE%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="Δεκαεξαδικό σύστημα αρίθμησης – Greek" lang="el" hreflang="el" data-title="Δεκαεξαδικό σύστημα αρίθμησης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_hexadecimal" title="Sistema hexadecimal – Spanish" lang="es" hreflang="es" data-title="Sistema hexadecimal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Deksesuma_nombrosistemo" title="Deksesuma nombrosistemo – Esperanto" lang="eo" hreflang="eo" data-title="Deksesuma nombrosistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki-sistema_hamaseitar" title="Zenbaki-sistema hamaseitar – Basque" lang="eu" hreflang="eu" data-title="Zenbaki-sistema hamaseitar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%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%B1%DB%B6" title="دستگاه اعداد پایه ۱۶ – Persian" lang="fa" hreflang="fa" data-title="دستگاه اعداد پایه ۱۶" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_hexad%C3%A9cimal" title="Système hexadécimal – French" lang="fr" hreflang="fr" data-title="Système hexadécimal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Heksadesimaal_systeem" title="Heksadesimaal systeem – Western Frisian" lang="fy" hreflang="fy" data-title="Heksadesimaal systeem" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%B3d%C3%BA_heicsidheach%C3%BAlach" title="Códú heicsidheachúlach – Irish" lang="ga" hreflang="ga" data-title="Códú heicsidheachúlach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/C%C3%B3digo_hexadecimal" title="Código hexadecimal – Galician" lang="gl" hreflang="gl" data-title="Código hexadecimal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%AD%EC%9C%A1%EC%A7%84%EB%B2%95" title="십육진법 – Korean" lang="ko" hreflang="ko" data-title="십육진법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</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%BF%D5%A1%D5%BD%D5%B6%D5%BE%D5%A5%D6%81%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 տասնվեցական համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Հաշվարկման տասնվեցական համակարգ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B7%E0%A5%8B%E0%A4%A1%E0%A4%B6_%E0%A4%86%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A5%80" title="षोडश आधारी – Hindi" lang="hi" hreflang="hi" data-title="षोडश आधारी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Heksadekadski_brojevni_sustav" title="Heksadekadski brojevni sustav – Croatian" lang="hr" hreflang="hr" data-title="Heksadekadski brojevni sustav" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Hexadecimala_nombrosistemo" title="Hexadecimala nombrosistemo – Ido" lang="io" hreflang="io" data-title="Hexadecimala nombrosistemo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Heksadesimal" title="Heksadesimal – Indonesian" lang="id" hreflang="id" data-title="Heksadesimal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sext%C3%A1nundakerfi" title="Sextánundakerfi – Icelandic" lang="is" hreflang="is" data-title="Sextánundakerfi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_numerico_esadecimale" title="Sistema numerico esadecimale – Italian" lang="it" hreflang="it" data-title="Sistema numerico esadecimale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%A1%D7%99%D7%A1_%D7%94%D7%A7%D7%A1%D7%93%D7%A6%D7%99%D7%9E%D7%9C%D7%99" title="בסיס הקסדצימלי – Hebrew" lang="he" hreflang="he" data-title="בסיס הקסדצימלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/H%C3%A8ksad%C3%A8simal" title="Hèksadèsimal – Javanese" lang="jv" hreflang="jv" data-title="Hèksadèsimal" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BD%D0%B0%D1%83%D0%B4%D1%8B%D2%A3_%D0%BE%D0%BD_%D0%B0%D0%BB%D1%82%D1%8B%D0%BB%D1%8B%D2%9B_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Санаудың он алтылық жүйесі – Kazakh" lang="kk" hreflang="kk" data-title="Санаудың он алтылық жүйесі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_ekzadesimal" title="Sistèm ekzadesimal – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm ekzadesimal" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9E%D0%BD_%D0%B0%D0%BB%D1%82%D1%8B%D0%BB%D1%8B%D0%BA_%D1%8D%D1%81%D0%B5%D0%BF%D1%82%D3%A9%D3%A9_%D1%82%D1%83%D1%82%D1%83%D0%BC%D1%83" title="Он алтылык эсептөө тутуму – Kyrgyz" lang="ky" hreflang="ky" data-title="Он алтылык эсептөө тутуму" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_numericum_sedecimale" title="Systema numericum sedecimale – Latin" lang="la" hreflang="la" data-title="Systema numericum sedecimale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Heksadecim%C4%81l%C4%81_skait%C4%AB%C5%A1anas_sist%C4%93ma" title="Heksadecimālā skaitīšanas sistēma – Latvian" lang="lv" hreflang="lv" data-title="Heksadecimālā skaitīšanas sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sistem_exadesimal" title="Sistem exadesimal – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Sistem exadesimal" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/ju%E2%80%99u_pi%E2%80%99e" title="ju’u pi’e – Lojban" lang="jbo" hreflang="jbo" data-title="ju’u pi’e" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Sistema_esadecimal" title="Sistema esadecimal – Lombard" lang="lmo" hreflang="lmo" data-title="Sistema esadecimal" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Tizenhatos_sz%C3%A1mrendszer" title="Tizenhatos számrendszer – Hungarian" lang="hu" hreflang="hu" data-title="Tizenhatos számrendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A8%D0%B5%D1%81%D0%BD%D0%B0%D0%B5%D1%81%D0%B5%D1%82%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="Шеснаесетеречен броен систем – Macedonian" lang="mk" hreflang="mk" data-title="Шеснаесетеречен броен систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B7%E0%B5%8B%E0%B4%A1%E0%B4%B6%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%B8%E0%B4%AE%E0%B5%8D%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A6%E0%B4%BE%E0%B4%AF%E0%B4%82" title="ഷോഡശസംഖ്യാസമ്പ്രദായം – Malayalam" lang="ml" hreflang="ml" data-title="ഷോഡശസംഖ്യാസമ്പ്രദായം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B7%E0%A5%8B%E0%A4%A1%E0%A4%B6%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="षोडशमान – Marathi" lang="mr" hreflang="mr" data-title="षोडशमान" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_perenambelasan" title="Nombor perenambelasan – Malay" lang="ms" hreflang="ms" data-title="Nombor perenambelasan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sistema_de_numera%C3%A7on_heixadecimal" title="Sistema de numeraçon heixadecimal – Mirandese" lang="mwl" hreflang="mwl" data-title="Sistema de numeraçon heixadecimal" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D1%80%D0%B2%D0%B0%D0%BD_%D0%B7%D1%83%D1%80%D0%B3%D0%B0%D0%B0%D1%82%D1%8B%D0%BD_%D1%82%D0%BE%D0%BE%D0%BB%D0%BB%D1%8B%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Арван зургаатын тооллын систем – Mongolian" lang="mn" hreflang="mn" data-title="Арван зургаатын тооллын систем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hexadecimaal_talstelsel" title="Hexadecimaal talstelsel – Dutch" lang="nl" hreflang="nl" data-title="Hexadecimaal talstelsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95" title="十六進法 – Japanese" lang="ja" hreflang="ja" data-title="十六進法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sekstentallsystemet" title="Sekstentallsystemet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sekstentallsystemet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sekstentalssystemet" title="Sekstentalssystemet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sekstentalssystemet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/O%CA%BBn_oltilik_sanoq_sistemasi" title="Oʻn oltilik sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Oʻn oltilik sanoq sistemasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szesnastkowy_system_liczbowy" title="Szesnastkowy system liczbowy – Polish" lang="pl" hreflang="pl" data-title="Szesnastkowy system liczbowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_numera%C3%A7%C3%A3o_hexadecimal" title="Sistema de numeração hexadecimal – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de numeração hexadecimal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_hexazecimal" title="Sistem hexazecimal – Romanian" lang="ro" hreflang="ro" data-title="Sistem hexazecimal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A8%D0%B5%D1%81%D1%82%D0%BD%D0%B0%D0%B4%D1%86%D0%B0%D1%82%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="Шестнадцатеричная система счисления – Russian" lang="ru" hreflang="ru" data-title="Шестнадцатеричная система счисления" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/Letlase_la_lesometshela" title="Letlase la lesometshela – Northern Sotho" lang="nso" hreflang="nso" data-title="Letlase la lesometshela" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Northern Sotho" class="interlanguage-link-target">Sesotho sa Leboa</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_i_numrave_heksadecimal" title="Sistemi i numrave heksadecimal – Albanian" lang="sq" hreflang="sq" data-title="Sistemi i numrave heksadecimal" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hexadecimal" title="Hexadecimal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Hexadecimal" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C5%A0estn%C3%A1stkov%C3%A1_s%C3%BAstava" title="Šestnástková sústava – Slovak" lang="sk" hreflang="sk" data-title="Šestnástková sústava" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/%C5%A0estnajsti%C5%A1ki_%C5%A1tevilski_sistem" title="Šestnajstiški številski sistem – Slovenian" lang="sl" hreflang="sl" data-title="Šestnajstiški številski sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B5%D0%BA%D1%81%D0%B0%D0%B4%D0%B5%D1%86%D0%B8%D0%BC%D0%B0%D0%BB%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Хексадецимални систем – Serbian" lang="sr" hreflang="sr" data-title="Хексадецимални систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Heksadecimalni_sistem" title="Heksadecimalni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Heksadecimalni sistem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Heksadesimaalij%C3%A4rjestelm%C3%A4" title="Heksadesimaalijärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Heksadesimaalijärjestelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hexadecimala_talsystemet" title="Hexadecimala talsystemet – Swedish" lang="sv" hreflang="sv" data-title="Hexadecimala talsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Hexadecimal" title="Hexadecimal – Tagalog" lang="tl" hreflang="tl" data-title="Hexadecimal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AE%B1%E0%AF%81%E0%AE%AE_%E0%AE%8E%E0%AE%A3%E0%AF%8D_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="பதினறும எண் முறைமை – Tamil" lang="ta" hreflang="ta" data-title="பதினறும எண் முறைமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%AA%E0%B8%B4%E0%B8%9A%E0%B8%AB%E0%B8%81" title="เลขฐานสิบหก – Thai" lang="th" hreflang="th" data-title="เลขฐานสิบหก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/On_alt%C4%B1l%C4%B1_say%C4%B1_sistemi" title="On altılı sayı sistemi – Turkish" lang="tr" hreflang="tr" data-title="On altılı sayı sistemi" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic</a></li> <li><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul> <hr /> <ul><li><a href="/wiki/Bengali_numerals" title="Bengali numerals">Bengali</a></li> <li><a href="/wiki/Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li> <li><a href="/wiki/Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li> <li><a href="/wiki/Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li> <li><a href="/wiki/Odia_numerals" title="Odia numerals">Odia</a></li> <li><a href="/wiki/Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li> <li><a href="/wiki/Tamil_numerals" title="Tamil numerals">Tamil</a></li> <li><a href="/wiki/Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li> <li><a href="/wiki/Telugu_script#Numerals" title="Telugu script">Telugu</a></li> <li><a href="/wiki/Kannada_script#Numerals" title="Kannada script">Kannada</a></li> <li><a href="/wiki/Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul> <hr /> <ul><li><a href="/wiki/Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li> <li><a href="/wiki/Balinese_numerals" title="Balinese numerals">Balinese</a></li> <li><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></li> <li><a href="/wiki/Javanese_numerals" title="Javanese numerals">Javanese</a></li> <li><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></li> <li><a href="/wiki/Lao_script#Numerals" title="Lao script">Lao</a></li> <li><a href="/wiki/Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li> <li><a href="/wiki/Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li> <li><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a> <ul><li><a href="/wiki/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 href="/wiki/Octal" title="Octal">8</a></li> <li><a href="/wiki/Decimal" title="Decimal">10</a></li> <li><a href="/wiki/Duodecimal" title="Duodecimal">12</a></li> <li><a class="mw-selflink selflink">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> (<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> (<a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li> <li><a href="/wiki/Mixed_radix" title="Mixed radix">Mixed</a> (<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> (<a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">2i</a>)</li> <li><a href="/wiki/Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a> (<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> Hexadecimal (also known as base-16 or simply hex) is a <a href="/wiki/Numeral_system#Positional_systems_in_detail" title="Numeral system">positional numeral system</a> that represents numbers using a <a href="/wiki/Radix" title="Radix">radix</a> (base) of sixteen. Unlike the <a href="/wiki/Decimal" title="Decimal">decimal</a> system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of <a href="/wiki/Binary_code" title="Binary code">binary-coded</a> values. Each hexadecimal digit represents four <a href="/wiki/Bit" title="Bit">bits</a> (binary digits), also known as a <a href="/wiki/Nibble" title="Nibble">nibble</a> (or nybble).<a href="#cite_note-1">[1]</a> For example, an 8-bit <a href="/wiki/Byte" title="Byte">byte</a> is two hexadecimal digits and its value can be written as <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style>00 to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value 711 would be expressed in hexadecimal as 2C716. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix <code>0x</code> is used in <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a>, which would denote this value as <code>0x2C7</code>. Hexadecimal is used in the transfer encoding Base 16, in which each byte of the <a href="/wiki/Plaintext" title="Plaintext">plain text</a> is broken into two 4-bit values and represented by two hexadecimal digits. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Representation">Representation</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=1" title="Edit section: Representation">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Written_representation">Written representation</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=2" title="Edit section: Written representation">edit</a>]</div> In most current use cases, the letters A–F or a–f represent the values 10–15, while the <a href="/wiki/Arabic_numerals" title="Arabic numerals">numerals</a> 0–9 are used to represent their decimal values. There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Some <a href="/wiki/Seven-segment_display" title="Seven-segment display">seven-segment displays</a> use mixed-case 'A b C d E F' to distinguish the digits A–F from one another and from 0–9. There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following <a href="/wiki/Hex_dump" title="Hex dump">hex dump</a>, each 8-bit <a href="/wiki/Byte" title="Byte">byte</a> is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number. <div class="nowrap"><div class="mw-highlight mw-highlight-lang-hexdump mw-content-ltr" dir="ltr"><pre>00000000 57 69 6b 69 70 65 64 69 61 2c 20 74 68 65 20 66 00000010 72 65 65 20 65 6e 63 79 63 6c 6f 70 65 64 69 61 00000020 20 74 68 61 74 20 61 6e 79 6f 6e 65 20 63 61 6e 00000030 20 65 64 69 74 0a </pre></div></div> <div class="mw-heading mw-heading4"><h4 id="Distinguishing_from_decimal">Distinguishing from decimal</h4>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=3" title="Edit section: Distinguishing from decimal">edit</a>]</div> In contexts where the <a href="/wiki/Radix" title="Radix">base</a> is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, which equals 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.<a href="#cite_note-2">[2]</a> Hexadecimal representations are written there in a <a href="/wiki/Monospaced_font" title="Monospaced font">typewriter typeface</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">5A3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">C1F27ED In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: <ul><li>Although best known from the <a href="/wiki/C_(programming_language)" title="C (programming language)">C programming language</a> (and the many languages influenced by C), the prefix <code>0x</code> to indicate a hex constant may have had origins in the <a href="/wiki/IBM_Stretch" class="mw-redirect" title="IBM Stretch">IBM Stretch</a> systems. It is derived from the <code>0</code> prefix already in use for <a href="/wiki/Octal" title="Octal">octal</a> constants. Byte values can be expressed in hexadecimal with the prefix <code>\x</code> followed by two hex digits: <code>'\x1B'</code> represents the <a href="/wiki/Escape_character" title="Escape character">Esc</a> control character; <code>"\x1B[0m\x1B[25;1H"</code> is a string containing 11 characters with two embedded Esc characters.<a href="#cite_note-3">[3]</a> To output an integer as hexadecimal with the <a href="/wiki/Printf" title="Printf">printf</a> function family, the format conversion code <code>%X</code> or <code>%x</code> is used.</li> <li>In <a href="/wiki/XML" title="XML">XML</a> and <a href="/wiki/XHTML" title="XHTML">XHTML</a>, characters can be expressed as hexadecimal <a href="/wiki/Numeric_character_reference" title="Numeric character reference">numeric character references</a> using the notation <code>&#xcode;</code>, for instance <code>&#x0054;</code> represents the character U+0054 (the uppercase letter "T"). If there is no <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">x</code> the number is decimal (thus <code>&#0084;</code> is the same character).<a href="#cite_note-4">[4]</a></li> <li>In Intel-derived <a href="/wiki/Assembly_language" title="Assembly language">assembly languages</a> and Modula-2,<a href="#cite_note-5">[5]</a> hexadecimal is denoted with a suffixed <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">H or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">h: <code>FFh</code> or <code>05A3H</code>. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write <code>0FFh</code> instead of <code>FFh</code>. Some other implementations (such as NASM) allow C-style numbers (<code>0x42</code>).</li> <li>Other assembly languages (<a href="/wiki/MOS_Technology_6502" title="MOS Technology 6502">6502</a>, <a href="/wiki/Motorola" title="Motorola">Motorola</a>), <a href="/wiki/Pascal_(programming_language)" title="Pascal (programming language)">Pascal</a>, <a href="/wiki/Delphi_(programming_language)" class="mw-redirect" title="Delphi (programming language)">Delphi</a>, some versions of <a href="/wiki/BASIC" title="BASIC">BASIC</a> (<a href="/wiki/Commodore_BASIC" title="Commodore BASIC">Commodore</a>), <a href="/wiki/GameMaker_Studio" class="mw-redirect" title="GameMaker Studio">GameMaker Language</a>, <a href="/wiki/Godot_(game_engine)" title="Godot (game engine)">Godot</a> and <a href="/wiki/Forth_(programming_language)" title="Forth (programming language)">Forth</a> use <code>$</code> as a prefix: <code>$5A3</code>, <code>$C1F27ED</code>.</li> <li>Some assembly languages (Microchip) use the notation <code>H'ABCD'</code> (for ABCD16). Similarly, <a href="/wiki/Fortran_95_language_features" title="Fortran 95 language features">Fortran 95</a> uses Z'ABCD'.</li> <li><a href="/wiki/Ada_(programming_language)" title="Ada (programming language)">Ada</a> and <a href="/wiki/VHDL" title="VHDL">VHDL</a> enclose hexadecimal numerals in based "numeric quotes": <code>16#5A3#</code>, <code>16#C1F27ED#</code>. For bit vector constants <a href="/wiki/VHDL" title="VHDL">VHDL</a> uses the notation <code>x"5A3"</code>, <code>x"C1F27ED"</code>.<a href="#cite_note-6">[6]</a></li> <li><a href="/wiki/Verilog" title="Verilog">Verilog</a> represents hexadecimal constants in the form <code>8'hFF</code>, where 8 is the number of bits in the value and FF is the hexadecimal constant.</li> <li>The <a href="/wiki/Icon_(programming_language)" title="Icon (programming language)">Icon</a> and <a href="/wiki/Smalltalk" title="Smalltalk">Smalltalk</a> languages use the prefix <code>16r</code>: <code>16r5A3</code></li> <li><a href="/wiki/PostScript" title="PostScript">PostScript</a> and the <a href="/wiki/Bourne_shell" title="Bourne shell">Bourne shell</a> and its derivatives denote hex with prefix <code>16#</code>: <code>16#5A3</code>, <code>16#C1F27ED</code>.</li> <li><a href="/wiki/Common_Lisp" title="Common Lisp">Common Lisp</a> uses the prefixes <code>#x</code> and <code>#16r</code>. Setting the variables *read-base*<a href="#cite_note-7">[7]</a> and *print-base*<a href="#cite_note-8">[8]</a> to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.</li> <li><a href="/wiki/MSX_BASIC" title="MSX BASIC">MSX BASIC</a>,<a href="#cite_note-9">[9]</a> <a href="/wiki/QuickBASIC" title="QuickBASIC">QuickBASIC</a>, <a href="/wiki/FreeBASIC" title="FreeBASIC">FreeBASIC</a> and <a href="/wiki/Visual_Basic" title="Visual Basic">Visual Basic</a> prefix hexadecimal numbers with <code>&H</code>: <code>&H5A3</code></li> <li><a href="/wiki/BBC_BASIC" title="BBC BASIC">BBC BASIC</a> and <a href="/wiki/Locomotive_BASIC" title="Locomotive BASIC">Locomotive BASIC</a> use <code>&</code> for hex.<a href="#cite_note-10">[10]</a></li> <li><a href="/wiki/TI-89" class="mw-redirect" title="TI-89">TI-89</a> and 92 series uses a <code>0h</code> prefix: <code>0h5A3</code>, <code>0hC1F27ED</code></li> <li><a href="/wiki/ALGOL_68" title="ALGOL 68">ALGOL 68</a> uses the prefix <code>16r</code> to denote hexadecimal numbers: <code>16r5a3</code>, <code>16rC1F27ED</code>. Binary, quaternary (base-4), and octal numbers can be specified similarly.</li> <li>The most common format for hexadecimal on IBM mainframes (<a href="/wiki/ZSeries" class="mw-redirect" title="ZSeries">zSeries</a>) and midrange computers (<a href="/wiki/IBM_i" title="IBM i">IBM i</a>) running the traditional OS's (<a href="/wiki/Z/OS" title="Z/OS">zOS</a>, <a href="/wiki/VSE_(operating_system)" title="VSE (operating system)">zVSE</a>, <a href="/wiki/Z/VM" title="Z/VM">zVM</a>, <a href="/wiki/Transaction_Processing_Facility" title="Transaction Processing Facility">TPF</a>, <a href="/wiki/IBM_i" title="IBM i">IBM i</a>) is <code>X'5A3'</code> or <code>X'C1F27ED'</code>, and is used in Assembler, <a href="/wiki/PL/I" title="PL/I">PL/I</a>, <a href="/wiki/COBOL" title="COBOL">COBOL</a>, <a href="/wiki/Job_Control_Language" title="Job Control Language">JCL</a>, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Syntax_that_is_always_Hex">Syntax that is always Hex</h4>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=4" title="Edit section: Syntax that is always Hex">edit</a>]</div> Sometimes the numbers are known to be Hex. <ul><li>In <a href="/wiki/URI" class="mw-redirect" title="URI">URIs</a> (including <a href="/wiki/URL" title="URL">URLs</a>), <a href="/wiki/Character_encoding" title="Character encoding">character codes</a> are written as hexadecimal pairs prefixed with <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">http://www.example.com/name%20with%20spaces</code> where <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%20</code> is the code for the <a href="/wiki/Space_(punctuation)#Space_characters_and_digital_typography" title="Space (punctuation)">space (blank)</a> character, <a href="/wiki/ASCII" title="ASCII">ASCII</a> code point 20 in hex, 32 in decimal.</li> <li>In the <a href="/wiki/Unicode" title="Unicode">Unicode</a> standard, a character value is represented with <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">U+</code> followed by the hex value, e.g. <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">U+00A1</code> is the <a href="/wiki/Inverted_exclamation_point" class="mw-redirect" title="Inverted exclamation point">inverted exclamation point</a> (¡).</li> <li><a href="/wiki/Web_colors" title="Web colors">Color references</a> in HTML, <a href="/wiki/Cascading_Style_Sheets" class="mw-redirect" title="Cascading Style Sheets">CSS</a> and <a href="/wiki/X_window_system" class="mw-redirect" title="X window system">X Window</a> can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">#</code>: magenta, for example, is represented as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">#FF00FF</code>.<a href="#cite_note-11">[11]</a> CSS also allows 3-hexdigit abbreviations with one hexdigit per component: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">#FA3</code> abbreviates <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">#FFAA33</code> (a golden orange: <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style> ).</li> <li>In <a href="/wiki/MIME" title="MIME">MIME</a> (e-mail extensions) <a href="/wiki/Quoted-printable" title="Quoted-printable">quoted-printable</a> encoding, character codes are written as hexadecimal pairs prefixed with <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">=</code>: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Espa=F1a</code> is "España" (F1hex is the code for ñ in the ISO/IEC 8859-1 character set).<a href="#cite_note-12">[12]</a>)</li> <li>PostScript binary data (such as image <a href="/wiki/Pixel" title="Pixel">pixels</a>) can be expressed as unprefixed consecutive hexadecimal pairs: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">AA213FD51B3801043FBC</code> ...</li> <li>Any <a href="/wiki/IPv6_address" title="IPv6 address">IPv6 address</a> can be written as eight groups of four hexadecimal digits (sometimes called <a href="/wiki/Hextet_(computing)" class="mw-redirect" title="Hextet (computing)">hextets</a>), where each group is separated by a colon (<code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">:</code>). This, for example, is a valid IPv6 address: <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">2001:0db8:85a3:0000:0000:8a2e:0370:7334</code> or abbreviated by removing leading zeros as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">2001:db8:85a3::8a2e:370:7334</code> (<a href="/wiki/IPv4_address" class="mw-redirect" title="IPv4 address">IPv4 addresses</a> are usually written in decimal).</li> <li><a href="/wiki/Globally_unique_identifier" class="mw-redirect" title="Globally unique identifier">Globally unique identifiers</a> are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">3F2504E0-4F89-41D3-9A0C-0305E82C3301</code>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_symbols_for_10–15_and_mostly_different_symbol_sets">Other symbols for 10–15 and mostly different symbol sets</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=5" title="Edit section: Other symbols for 10–15 and mostly different symbol sets">edit</a>]</div> The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers. <ul><li>During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an <a href="/wiki/Overline" title="Overline">overline</a> to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.</li> <li>The <a href="/wiki/SWAC_(computer)" title="SWAC (computer)">SWAC</a> (1950)<a href="#cite_note-Savard_2018_CA-13">[13]</a> and <a href="/wiki/Bendix_G-15" title="Bendix G-15">Bendix G-15</a> (1956)<a href="#cite_note-Bendix-14">[14]</a><a href="#cite_note-Savard_2018_CA-13">[13]</a> computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.</li> <li>The <a href="/wiki/ORDVAC" title="ORDVAC">ORDVAC</a> and <a href="/wiki/ILLIAC_I" title="ILLIAC I">ILLIAC I</a> (1952) computers (and some derived designs, e.g. <a href="/wiki/BRLESC" title="BRLESC">BRLESC</a>) used the uppercase letters K, S, N, J, F and L for the values 10 to 15.<a href="#cite_note-Illiac-I-15">[15]</a><a href="#cite_note-Savard_2018_CA-13">[13]</a></li> <li>The Librascope <a href="/wiki/LGP-30" title="LGP-30">LGP-30</a> (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.<a href="#cite_note-RP_1957_LGP-30-16">[16]</a><a href="#cite_note-Savard_2018_CA-13">[13]</a></li> <li>On the <a href="/wiki/PERM_(computer)" title="PERM (computer)">PERM</a> (1956) computer, hexadecimal numbers were written as letters O for zero, A to N and P for 1 to 15. Many machine instructions had mnemonic hex-codes (A=add, M=multiply, L=load, F=fixed-point etc.); programs were written without instruction names.<a href="#cite_note-PERM-17">[17]</a></li> <li>The <a href="/wiki/Honeywell" title="Honeywell">Honeywell</a> <a href="/wiki/Datamatic_D-1000" class="mw-redirect" title="Datamatic D-1000">Datamatic D-1000</a> (1957) used the lowercase letters b, c, d, e, f, and g whereas the <a href="/wiki/Elbit" class="mw-redirect" title="Elbit">Elbit</a> 100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.<a href="#cite_note-Savard_2018_CA-13">[13]</a></li> <li>The <a href="/wiki/Monrobot_XI" title="Monrobot XI">Monrobot XI</a> (1960) used the letters S, T, U, V, W and X for the values 10 to 15.<a href="#cite_note-Savard_2018_CA-13">[13]</a></li> <li>The <a href="/wiki/NEC" title="NEC">NEC</a> <a href="/wiki/Parametron" title="Parametron">parametron</a> computer NEAC 1103 (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.<a href="#cite_note-NEC_1960_NEAC-1103-18">[18]</a></li> <li>The <a href="/wiki/Pacific_Data_Systems" class="mw-redirect" title="Pacific Data Systems">Pacific Data Systems</a> 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.<a href="#cite_note-Savard_2018_CA-13">[13]</a></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg/220px-Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg.png" decoding="async" width="220" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg/330px-Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg/440px-Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg.png 2x" data-file-width="890" data-file-height="141" /></a><figcaption><a href="/wiki/Bibi-binary" title="Bibi-binary">Bibi-binary</a></figcaption></figure> <ul><li>New numeric symbols and names were introduced in the <a href="/wiki/Bibi-binary" title="Bibi-binary">Bibi-binary</a> notation by <a href="/wiki/Boby_Lapointe" title="Boby Lapointe">Boby Lapointe</a> in 1968.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bruce_Martin_hexadecimal_notation_proposal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Bruce_Martin_hexadecimal_notation_proposal.png/220px-Bruce_Martin_hexadecimal_notation_proposal.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Bruce_Martin_hexadecimal_notation_proposal.png/330px-Bruce_Martin_hexadecimal_notation_proposal.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Bruce_Martin_hexadecimal_notation_proposal.png/440px-Bruce_Martin_hexadecimal_notation_proposal.png 2x" data-file-width="787" data-file-height="545" /></a><figcaption>Bruce Alan Martin's hexadecimal notation proposal<a href="#cite_note-Martin_1968-19">[19]</a></figcaption></figure> <ul><li>Bruce Alan Martin of <a href="/wiki/Brookhaven_National_Laboratory" title="Brookhaven National Laboratory">Brookhaven National Laboratory</a> considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">CACM</a>, he proposed an entirely new set of symbols based on the bit locations.<a href="#cite_note-Martin_1968-19">[19]</a></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Base-16_digits.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Base-16_digits.svg/220px-Base-16_digits.svg.png" decoding="async" width="220" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Base-16_digits.svg/330px-Base-16_digits.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Base-16_digits.svg/440px-Base-16_digits.svg.png 2x" data-file-width="945" data-file-height="495" /></a><figcaption>Ronald O. Whitaker's hexadecimal notation proposal.<a href="#cite_note-Whitaker_1972-20">[20]</a><a href="#cite_note-Whitaker_1975-21">[21]</a></figcaption></figure> <ul><li>In 1972, Ronald O. Whitaker of Rowco Engineering Co. proposed a triangular font that allows "direct binary reading" to "permit both input and output from computers without respect to encoding matrices."<a href="#cite_note-Whitaker_1972-20">[20]</a><a href="#cite_note-Whitaker_1975-21">[21]</a></li> <li>Some <a href="/wiki/Seven-segment_display" title="Seven-segment display">seven-segment display</a> decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.<a href="#cite_note-22">[22]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Verbal_and_digital_representations">Verbal and digital representations</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=6" title="Edit section: Verbal and digital representations">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hexadecimal-counting.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hexadecimal-counting.jpg/220px-Hexadecimal-counting.jpg" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hexadecimal-counting.jpg/330px-Hexadecimal-counting.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b5/Hexadecimal-counting.jpg 2x" data-file-width="412" data-file-height="331" /></a><figcaption>Hexadecimal finger-counting scheme</figcaption></figure> Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the <a href="/wiki/ICAO_spelling_alphabet" class="mw-redirect" title="ICAO spelling alphabet">NATO phonetic alphabet</a>, the <a href="/wiki/Joint_Army/Navy_Phonetic_Alphabet" class="mw-redirect" title="Joint Army/Navy Phonetic Alphabet">Joint Army/Navy Phonetic Alphabet</a>, or a similar ad-hoc system. In the wake of the adoption of hexadecimal among <a href="/wiki/IBM_System/360" title="IBM System/360">IBM System/360</a> programmers, Magnuson (1968)<a href="#cite_note-Magnuson-1968-01-23">[23]</a> suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.<a href="#cite_note-Magnuson-1968-01-23">[23]</a> Another naming-system was published online by Rogers (2007)<a href="#cite_note-Rogers-2007-24">[24]</a> that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in <a href="/wiki/Silicon_Valley_(TV_series)" title="Silicon Valley (TV series)">Silicon Valley</a>.<a href="#cite_note-Babb-2015-25">[25]</a> The system proposed by Babb was further improved by Atkins-Bittner in 2015-2016.<a href="#cite_note-Atkins-Bittner_2015-26">[26]</a> Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15. Systems of counting on <a href="/wiki/Digit_(anatomy)" title="Digit (anatomy)">digits</a> have been devised for both binary and hexadecimal. <a href="/wiki/Arthur_C._Clarke" title="Arthur C. Clarke">Arthur C. Clarke</a> suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers.<a href="#cite_note-27">[27]</a> Another system for counting up to FF16 (25510) is illustrated on the right. <table class="wikitable" style="display: inline-table; margin-right: 50px;; text-align:right;"> <caption>Magnuson (1968)<a href="#cite_note-Magnuson-1968-01-23">[23]</a> naming method </caption> <tbody><tr> <th>Number</th> <th>Pronunciation </th></tr> <tr> <td>A</td> <td>ann </td></tr> <tr> <td>B</td> <td>bet </td></tr> <tr> <td>C</td> <td>chris </td></tr> <tr> <td>D</td> <td>dot </td></tr> <tr> <td>E</td> <td>ernest </td></tr> <tr> <td>F</td> <td>frost </td></tr> <tr> <td>1A</td> <td>annteen </td></tr> <tr> <td>A0</td> <td>annty </td></tr> <tr> <td>5B</td> <td>fifty-bet </td></tr> <tr> <td>A,01C</td> <td>annty christeen </td></tr> <tr> <td>1,AD0</td> <td>annteen dotty </td></tr> <tr> <td>3,A7D</td> <td>thirty-ann seventy-dot </td></tr></tbody></table> <table class="wikitable" style="display: inline-table; margin-right: 50px;; text-align:right;"> <caption>Rogers (2007)<a href="#cite_note-Rogers-2007-24">[24]</a> naming method </caption> <tbody><tr> <th>Number</th> <th>Pronunciation </th></tr> <tr> <td>A</td> <td>ten </td></tr> <tr> <td>B</td> <td>eleven </td></tr> <tr> <td>C</td> <td>twelve </td></tr> <tr> <td>D</td> <td>draze </td></tr> <tr> <td>E</td> <td>eptwin </td></tr> <tr> <td>F</td> <td>fim </td></tr> <tr> <td>10</td> <td>tex </td></tr> <tr> <td>11</td> <td>oneteek </td></tr> <tr> <td>1F</td> <td>fimteek </td></tr> <tr> <td>50</td> <td>fiftek </td></tr> <tr> <td>C0</td> <td>twelftek </td></tr> <tr> <td>100</td> <td>hundrek </td></tr> <tr> <td>1,000</td> <td>thousek </td></tr> <tr> <td>3E</td> <td>thirtek-eptwin </td></tr> <tr> <td>E1</td> <td>eptek-one </td></tr> <tr> <td>C4A</td> <td>twelve-hundrek-fourtek-ten </td></tr> <tr> <td>1,743</td> <td>one-thousek-seven- -hundrek-fourtek-three </td></tr></tbody></table> <table class="wikitable" style="display: inline-table; margin-right: 50px;; text-align:right;"> <caption>Atkins-Bittner (2015)<a href="#cite_note-Atkins-Bittner_2015-26">[26]</a> naming method </caption> <tbody><tr style="font-weight:bold; text-align:center;"> <th>Number </th> <th>Pronunciation </th></tr> <tr> <td>A </td> <td>ae </td></tr> <tr> <td>B </td> <td>bee </td></tr> <tr> <td>C </td> <td>cee </td></tr> <tr> <td>D </td> <td>dee </td></tr> <tr> <td>E </td> <td>ee </td></tr> <tr> <td>F </td> <td>eff </td></tr> <tr> <td>A0 </td> <td>atta </td></tr> <tr> <td>B0 </td> <td>bitta </td></tr> <tr> <td>C0 </td> <td>citta </td></tr> <tr> <td>D0 </td> <td>dickety </td></tr> <tr> <td>E0 </td> <td>eckity </td></tr> <tr> <td>F0 </td> <td>fleventy </td></tr> <tr> <td>1A </td> <td>abteen </td></tr> <tr> <td>1B </td> <td>bibteen </td></tr> <tr> <td>1C </td> <td>cibteen </td></tr> <tr> <td>1D </td> <td>dibbleteen </td></tr> <tr> <td>1E </td> <td>ebbleteen </td></tr> <tr> <td>1F </td> <td>fleventeen </td></tr> <tr> <td>100 </td> <td>one bitey </td></tr> <tr> <td>10,000 </td> <td>one millby </td></tr> <tr> <td>100,000,000 </td> <td>one billby </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Signs">Signs</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=7" title="Edit section: Signs">edit</a>]</div> The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210, −B01D9 to represent −72136910 and so on. Hexadecimal can also be used to express the exact bit patterns used in the <a href="/wiki/Central_processing_unit" title="Central processing unit">processor</a>, so a sequence of hexadecimal digits may represent a <a href="/wiki/Signedness" title="Signedness">signed</a> or even a <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point</a> value. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bit <a href="/wiki/Processor_register" title="Processor register">CPU register</a> (in <a href="/wiki/Two%27s_complement" title="Two's complement">two's complement</a>), as C228 0000 in a 32-bit <a href="/wiki/Floating-point_unit" title="Floating-point unit">FPU</a> register or C045 0000 0000 0000 in a 64-bit FPU register (in the <a href="/wiki/IEEE_floating-point_standard" class="mw-redirect" title="IEEE floating-point standard">IEEE floating-point standard</a>). <div class="mw-heading mw-heading3"><h3 id="Hexadecimal_exponential_notation">Hexadecimal exponential notation</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=8" title="Edit section: Hexadecimal exponential notation">edit</a>]</div> Just as decimal numbers can be represented in <a href="/wiki/Exponential_notation" class="mw-redirect" title="Exponential notation">exponential notation</a>, so too can hexadecimal numbers. <a href="/wiki/P_notation" class="mw-redirect" title="P notation">P notation</a> uses the letter P (or p, for "power"), whereas E (or e) serves a similar purpose in decimal <a href="/wiki/E_notation" class="mw-redirect" title="E notation">E notation</a>. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1p5. Usually, the number is normalized so that the hexadecimal digits start with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">1. (zero is usually <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">0 with no P). Example: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">1.3DEp42 represents 1.3DE16 × 24210. P notation is required by the <a href="/wiki/IEEE_754-2008" class="mw-redirect" title="IEEE 754-2008">IEEE 754-2008</a> binary floating-point standard and can be used for floating-point literals in the <a href="/wiki/C99" title="C99">C99</a> edition of the <a href="/wiki/C_(programming_language)" title="C (programming language)">C programming language</a>.<a href="#cite_note-28">[28]</a> Using the %a or %A conversion specifiers, this notation can be produced by implementations of the <a href="/wiki/Printf" title="Printf">printf</a> family of functions following the C99 specification<a href="#cite_note-Rationale_2003_C-29">[29]</a> and <a href="/wiki/Single_UNIX_Specification" title="Single UNIX Specification">Single Unix Specification</a> (IEEE Std 1003.1) <a href="/wiki/POSIX" title="POSIX">POSIX</a> standard.<a href="#cite_note-printf_2013-30">[30]</a> <div class="mw-heading mw-heading2"><h2 id="Conversion">Conversion</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=9" title="Edit section: Conversion">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Binary_conversion">Binary conversion</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=10" title="Edit section: Binary conversion">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator,_HP%27s_First_and_Only_Calculator_esp._for_Programmers,_built_1982-1989_(edited_to_rectangular,_V2).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg/220px-Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg" decoding="async" width="220" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg/330px-Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg/440px-Hewlett-Packard_Model_HP-16C_Programmable_RPN_Calculator%2C_HP%27s_First_and_Only_Calculator_esp._for_Programmers%2C_built_1982-1989_%28edited_to_rectangular%2C_V2%29.jpg 2x" data-file-width="2449" data-file-height="1548" /></a><figcaption>The programmable <a href="/wiki/Reverse_Polish_notation" title="Reverse Polish notation">RPN</a>-calculator <a href="/wiki/HP-16C" title="HP-16C">HP-16C Computer Scientist</a> from 1982 was designed for programmers. One of its key features was the conversion between different numeral systems (note hex number in display).</figcaption></figure> Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten. Since each <a href="/wiki/Positional_notation" title="Positional notation">position</a> in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right: <ul><li>00012 = 110</li> <li>00102 = 210</li> <li>01002 = 410</li> <li>10002 = 810</li></ul> Therefore: <table> <tbody><tr> <td>11112</td> <td>= 810 + 410 + 210 + 110 </td></tr> <tr> <td> </td> <td>= 1510 </td></tr></tbody></table> With little practice, mapping 11112 to F16 in one step becomes easy (see table in <a href="#Written_representation">written representation</a>). The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.<a href="#cite_note-Mano-Ciletti-31">[31]</a> This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results. <table> <tbody><tr> <td>(1001011100)2</td> <td>= 51210 + 6410 + 1610 + 810 + 410 </td></tr> <tr> <td> </td> <td>= 60410 </td></tr></tbody></table> Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently and converted directly: <table> <tbody><tr> <td>(1001011100)2</td> <td>=</td> <td>0010 </td> <td>0101 </td> <td>11002 </td></tr> <tr> <td> </td> <td>=</td> <td align="center">2</td> <td align="center">5</td> <td align="center">C16 </td></tr> <tr> <td> </td> <td>=</td> <td colspan="5">25C16 </td></tr></tbody></table> The conversion from hexadecimal to binary is equally direct.<a href="#cite_note-Mano-Ciletti-31">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Other_simple_conversions">Other simple conversions</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=11" title="Edit section: Other simple conversions">edit</a>]</div> Although <a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">quaternary</a> (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C16 = 02 11 304. The <a href="/wiki/Octal" title="Octal">octal</a> (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four. <div class="mw-heading mw-heading3"><h3 id="Division-remainder_in_source_base">Division-remainder in source base</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=12" title="Edit section: Division-remainder in source base">edit</a>]</div> As with all bases there is a simple <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method. Let d be the number to represent in hexadecimal, and the series hihi−1...h2h1 be the hexadecimal digits representing the number. <ol><li>i ← 1</li> <li>hi ← d mod 16</li> <li>d ← (d − hi) / 16</li> <li>If d = 0 (return series hi) else increment i and go to step 2</li></ol> "16" may be replaced with any other base that may be desired. The following is a <a href="/wiki/JavaScript" title="JavaScript">JavaScript</a> implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with <a href="/wiki/Bitwise_operators" class="mw-redirect" title="Bitwise operators">bitwise operators</a>. <div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre>function toHex(d) { var r = d % 16; if (d - r == 0) { return toChar(r); } return toHex((d - r) / 16) + toChar(r); } function toChar(n) { const alpha = "0123456789ABCDEF"; return alpha.charAt(n); } </pre></div> <div class="mw-heading mw-heading3"><h3 id="Conversion_through_addition_and_multiplication">Conversion through addition and multiplication</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=13" title="Edit section: Conversion through addition and multiplication">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hexadecimal_multiplication_table.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hexadecimal_multiplication_table.svg/220px-Hexadecimal_multiplication_table.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hexadecimal_multiplication_table.svg/330px-Hexadecimal_multiplication_table.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hexadecimal_multiplication_table.svg/440px-Hexadecimal_multiplication_table.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>A hexadecimal <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a></figcaption></figure> It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that: B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160) which is 45997 in base 10. <div class="mw-heading mw-heading3"><h3 id="Tools_for_conversion">Tools for conversion</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=14" title="Edit section: Tools for conversion">edit</a>]</div> Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal. In <a href="/wiki/Microsoft_Windows" title="Microsoft Windows">Microsoft Windows</a>, the <a href="/wiki/Calculator_(Windows)" class="mw-redirect" title="Calculator (Windows)">Calculator</a>, on its Programmer mode, allows conversions between hexadecimal and other common programming bases. <div class="mw-heading mw-heading2"><h2 id="Elementary_arithmetic">Elementary arithmetic</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=15" title="Edit section: Elementary arithmetic">edit</a>]</div> Elementary operations such as division can be carried out indirectly through conversion to an alternate <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a>, such as the commonly used decimal system or the binary system where each hex digit corresponds to four binary digits. Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as <a href="/wiki/Long_division" title="Long division">long division</a> and the traditional subtraction algorithm. <div class="mw-heading mw-heading2"><h2 id="Real_numbers">Real numbers</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=16" title="Edit section: Real numbers">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Rational_numbers">Rational numbers</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=17" title="Edit section: Rational numbers">edit</a>]</div> As with other numeral systems, the hexadecimal system can be used to represent <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, although <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating expansions</a> are common since sixteen (1016) has only a single prime factor: two. For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> or dividing one by sixteen for hexadecimal, both of these fractions are written as <code>0.1</code>. Because the radix 16 is a <a href="/wiki/Square_number" title="Square number">perfect square</a> (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no <a href="/wiki/Cyclic_number" title="Cyclic number">cyclic numbers</a> (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a <a href="/wiki/Power_of_two" title="Power of two">power of two</a> result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than <a href="/wiki/Decimal" title="Decimal">decimal</a> for representing rational numbers since a larger proportion lies outside its range of finite representation. All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, <a href="/wiki/Duodecimal" title="Duodecimal">duodecimal</a> and <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a>: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560. <table class="wikitable"> <tbody><tr> <th rowspan="2" style="vertical-align:bottom;">n </th> <th colspan="3">Decimal Prime factors of: base, b = 10: <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5; b − 1 = 9: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3; b + 1 = 11: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </th> <th colspan="3">Hexadecimal Prime factors of: base, b = 1610 = 10: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2; b − 1 = 1510 = F: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 5; b + 1 = 1710 = 11: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </th></tr> <tr> <th>Reciprocal </th> <th>Prime factors </th> <th>Positional representation (decimal) </th> <th>Positional representation (hexadecimal) </th> <th>Prime factors </th> <th>Reciprocal </th></tr> <tr> <td>2 </td> <td align="center">1/2 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td>0.5 </td> <td>0.8 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td align="center">1/2 </td></tr> <tr> <td>3 </td> <td align="center">1/3 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.3333... = 0.3 </td> <td bgcolor="#e4e4e4">0.5555... = 0.5 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/3 </td></tr> <tr> <td>4 </td> <td align="center">1/4 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td>0.25 </td> <td>0.4 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td align="center">1/4 </td></tr> <tr> <td>5 </td> <td align="center">1/5 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td>0.2 </td> <td bgcolor="#e4e4e4">0.3 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/5 </td></tr> <tr> <td>6 </td> <td align="center">1/6 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.16 </td> <td bgcolor="#e4e4e4">0.2A </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/6 </td></tr> <tr> <td>7 </td> <td align="center">1/7 </td> <td align="center">7 </td> <td bgcolor="#e4e4e4">0.142857 </td> <td bgcolor="#e4e4e4">0.249 </td> <td align="center">7 </td> <td align="center">1/7 </td></tr> <tr> <td>8 </td> <td align="center">1/8 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td>0.125 </td> <td>0.2 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td align="center">1/8 </td></tr> <tr> <td>9 </td> <td align="center">1/9 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.1 </td> <td bgcolor="#e4e4e4">0.1C7 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/9 </td></tr> <tr> <td>10 </td> <td align="center">1/10 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td>0.1 </td> <td bgcolor="#e4e4e4">0.19 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/A </td></tr> <tr> <td>11 </td> <td align="center">1/11 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td bgcolor="#e4e4e4">0.09 </td> <td bgcolor="#e4e4e4">0.1745D </td> <td align="center">B </td> <td align="center">1/B </td></tr> <tr> <td>12 </td> <td align="center">1/12 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.083 </td> <td bgcolor="#e4e4e4">0.15 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/C </td></tr> <tr> <td>13 </td> <td align="center">1/13 </td> <td align="center">13 </td> <td bgcolor="#e4e4e4">0.076923 </td> <td bgcolor="#e4e4e4">0.13B </td> <td align="center">D </td> <td align="center">1/D </td></tr> <tr> <td>14 </td> <td align="center">1/14 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 7 </td> <td bgcolor="#e4e4e4">0.0714285 </td> <td bgcolor="#e4e4e4">0.1249 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 7 </td> <td align="center">1/E </td></tr> <tr> <td>15 </td> <td align="center">1/15 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td bgcolor="#e4e4e4">0.06 </td> <td bgcolor="#e4e4e4">0.1 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/F </td></tr> <tr> <td>16 </td> <td align="center">1/16 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td>0.0625 </td> <td>0.1 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td align="center">1/10 </td></tr> <tr> <td>17 </td> <td align="center">1/17 </td> <td align="center">17 </td> <td bgcolor="#e4e4e4">0.0588235294117647 </td> <td bgcolor="#e4e4e4">0.0F </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td align="center">1/11 </td></tr> <tr> <td>18 </td> <td align="center">1/18 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.05 </td> <td bgcolor="#e4e4e4">0.0E38 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/12 </td></tr> <tr> <td>19 </td> <td align="center">1/19 </td> <td align="center">19 </td> <td bgcolor="#e4e4e4">0.052631578947368421 </td> <td bgcolor="#e4e4e4">0.0D79435E5 </td> <td align="center">13 </td> <td align="center">1/13 </td></tr> <tr> <td>20 </td> <td align="center">1/20 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td>0.05 </td> <td bgcolor="#e4e4e4">0.0C </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/14 </td></tr> <tr> <td>21 </td> <td align="center">1/21 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 7 </td> <td bgcolor="#e4e4e4">0.047619 </td> <td bgcolor="#e4e4e4">0.0C3 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 7 </td> <td align="center">1/15 </td></tr> <tr> <td>22 </td> <td align="center">1/22 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td bgcolor="#e4e4e4">0.045 </td> <td bgcolor="#e4e4e4">0.0BA2E8 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, B </td> <td align="center">1/16 </td></tr> <tr> <td>23 </td> <td align="center">1/23 </td> <td align="center">23 </td> <td bgcolor="#e4e4e4">0.0434782608695652173913 </td> <td bgcolor="#e4e4e4">0.0B21642C859 </td> <td align="center">17 </td> <td align="center">1/17 </td></tr> <tr> <td>24 </td> <td align="center">1/24 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.0416 </td> <td bgcolor="#e4e4e4">0.0A </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/18 </td></tr> <tr> <td>25 </td> <td align="center">1/25 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td>0.04 </td> <td bgcolor="#e4e4e4">0.0A3D7 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/19 </td></tr> <tr> <td>26 </td> <td align="center">1/26 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 13 </td> <td bgcolor="#e4e4e4">0.0384615 </td> <td bgcolor="#e4e4e4">0.09D8 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, D </td> <td align="center">1/1A </td></tr> <tr> <td>27 </td> <td align="center">1/27 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.037 </td> <td bgcolor="#e4e4e4">0.097B425ED </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/1B </td></tr> <tr> <td>28 </td> <td align="center">1/28 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 7 </td> <td bgcolor="#e4e4e4">0.03571428 </td> <td bgcolor="#e4e4e4">0.0924 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 7 </td> <td align="center">1/1C </td></tr> <tr> <td>29 </td> <td align="center">1/29 </td> <td align="center">29 </td> <td bgcolor="#e4e4e4">0.0344827586206896551724137931 </td> <td bgcolor="#e4e4e4">0.08D3DCB </td> <td align="center">1D </td> <td align="center">1/1D </td></tr> <tr> <td>30 </td> <td align="center">1/30 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td bgcolor="#e4e4e4">0.03 </td> <td bgcolor="#e4e4e4">0.08 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/1E </td></tr> <tr> <td>31 </td> <td align="center">1/31 </td> <td align="center">31 </td> <td bgcolor="#e4e4e4">0.032258064516129 </td> <td bgcolor="#e4e4e4">0.08421 </td> <td align="center">1F </td> <td align="center">1/1F </td></tr> <tr> <td>32 </td> <td align="center">1/32 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td>0.03125 </td> <td>0.08 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2 </td> <td align="center">1/20 </td></tr> <tr> <td>33 </td> <td align="center">1/33 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td bgcolor="#e4e4e4">0.03 </td> <td bgcolor="#e4e4e4">0.07C1F </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, B </td> <td align="center">1/21 </td></tr> <tr> <td>34 </td> <td align="center">1/34 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 17 </td> <td bgcolor="#e4e4e4">0.02941176470588235 </td> <td bgcolor="#e4e4e4">0.078 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td align="center">1/22 </td></tr> <tr> <td>35 </td> <td align="center">1/35 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5, 7 </td> <td bgcolor="#e4e4e4">0.0285714 </td> <td bgcolor="#e4e4e4">0.075 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5, 7 </td> <td align="center">1/23 </td></tr> <tr> <td>36 </td> <td align="center">1/36 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.027 </td> <td bgcolor="#e4e4e4">0.071C </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/24 </td></tr> <tr> <td>37 </td> <td align="center">1/37 </td> <td align="center">37 </td> <td bgcolor="#e4e4e4">0.027 </td> <td bgcolor="#e4e4e4">0.06EB3E453 </td> <td align="center">25 </td> <td align="center">1/25 </td></tr> <tr> <td>38 </td> <td align="center">1/38 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 19 </td> <td bgcolor="#e4e4e4">0.02631578947368421 </td> <td bgcolor="#e4e4e4">0.0435E50D79435E4AC62B4 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 13 </td> <td align="center">1/26 </td></tr> <tr> <td>39 </td> <td align="center">1/39 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 13 </td> <td bgcolor="#e4e4e4">0.0256410 </td> <td bgcolor="#e4e4e4">0.069 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, D </td> <td align="center">1/27 </td></tr> <tr> <td>40 </td> <td align="center">1/40 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td>0.025 </td> <td bgcolor="#e4e4e4">0.06 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td align="center">1/28 </td></tr> <tr> <td>41 </td> <td align="center">1/41 </td> <td align="center">41 </td> <td bgcolor="#e4e4e4">0.02439 </td> <td bgcolor="#e4e4e4">0.063E9538D283B5B62FB8 </td> <td align="center">29 </td> <td align="center">1/29 </td></tr> <tr> <td>42 </td> <td align="center">1/42 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 7 </td> <td bgcolor="#e4e4e4">0.0238095 </td> <td bgcolor="#e4e4e4">0.0618 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 7 </td> <td align="center">1/2A </td></tr> <tr> <td>43 </td> <td align="center">1/43 </td> <td align="center">43 </td> <td bgcolor="#e4e4e4">0.023255813953488372093 </td> <td bgcolor="#e4e4e4">0.05F417D </td> <td align="center">2B </td> <td align="center">1/2B </td></tr> <tr> <td>44 </td> <td align="center">1/44 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">11 </td> <td bgcolor="#e4e4e4">0.0227 </td> <td bgcolor="#e4e4e4">0.05D1745 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, B </td> <td align="center">1/2C </td></tr> <tr> <td>45 </td> <td align="center">1/45 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">5 </td> <td bgcolor="#e4e4e4">0.02 </td> <td bgcolor="#e4e4e4">0.05B </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3, 5 </td> <td align="center">1/2D </td></tr> <tr> <td>46 </td> <td align="center">1/46 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 23 </td> <td bgcolor="#e4e4e4">0.02173913043478765869567 </td> <td bgcolor="#e4e4e4">0.0590B21642C9C4EF44A9 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, 17 </td> <td align="center">1/2E </td></tr> <tr> <td>47 </td> <td align="center">1/47 </td> <td align="center">47 </td> <td bgcolor="#e4e4e4">0.0212765957446808510638297872340425531914893617 </td> <td bgcolor="#e4e4e4">0.0572620AE4C415C97616<a href="#cite_note-32">[32]</a> </td> <td align="center">2F </td> <td align="center">1/2F </td></tr> <tr> <td>48 </td> <td align="center">1/48 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td bgcolor="#e4e4e4">0.02083 </td> <td bgcolor="#e4e4e4">0.05 </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">3 </td> <td align="center">1/30 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Irrational_numbers">Irrational numbers</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=18" title="Edit section: Irrational numbers">edit</a>]</div> The table below gives the expansions of some common <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> in decimal and hexadecimal. <table class="wikitable"> <tbody><tr> <th rowspan="2">Number </th> <th colspan="2">Positional representation </th></tr> <tr> <th>Decimal </th> <th>Hexadecimal </th></tr> <tr> <td><a href="/wiki/Square_root_of_2" title="Square root of 2">√2</a> (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>) </td> <td>1.414213562373095048... </td> <td>1.6A09E667F3BCD... </td></tr> <tr> <td><a href="/wiki/Square_root_of_3" title="Square root of 3">√3</a> (the length of the diagonal of a unit <a href="/wiki/Cube" title="Cube">cube</a>) </td> <td>1.732050807568877293... </td> <td>1.BB67AE8584CAA... </td></tr> <tr> <td><a href="/wiki/Square_root_of_5" title="Square root of 5">√5</a> (the length of the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> of a 1×2 <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>) </td> <td>2.236067977499789696... </td> <td>2.3C6EF372FE95... </td></tr> <tr> <td><a href="/wiki/Golden_ratio" title="Golden ratio">φ</a> (phi, the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> = (1+√5)/2) </td> <td>1.618033988749894848... </td> <td>1.9E3779B97F4A... </td></tr> <tr> <td><a href="/wiki/Pi" title="Pi">π</a> (pi, the ratio of <a href="/wiki/Circumference" title="Circumference">circumference</a> to <a href="/wiki/Diameter" title="Diameter">diameter</a> of a circle) </td> <td>3.141592653589793238462643 383279502884197169399375105... </td> <td>3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... </td></tr> <tr> <td><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> (the base of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>) </td> <td>2.718281828459045235... </td> <td>2.B7E151628AED2A6B... </td></tr> <tr> <td><a href="/wiki/Thue%E2%80%93Morse_constant" class="mw-redirect" title="Thue–Morse constant">τ</a> (the <a href="/wiki/Thue%E2%80%93Morse_constant" class="mw-redirect" title="Thue–Morse constant">Thue–Morse constant</a>) </td> <td>0.412454033640107597... </td> <td>0.6996 9669 9669 6996... </td></tr> <tr> <td><a href="/wiki/Euler-Mascheroni_constant" class="mw-redirect" title="Euler-Mascheroni constant">γ</a> (the limiting difference between the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a> and the natural logarithm) </td> <td>0.577215664901532860... </td> <td>0.93C467E37DB0C7A4D1B... </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Powers">Powers</h3>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=19" title="Edit section: Powers">edit</a>]</div> Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below. <table class="wikitable"> <tbody><tr> <th>2x</th> <th>Value</th> <th>Value (Decimal) </th></tr> <tr> <td>20</td> <td style="text-align:right;">1</td> <td style="text-align:right;">1 </td></tr> <tr> <td>21</td> <td style="text-align:right;">2</td> <td style="text-align:right;">2 </td></tr> <tr> <td>22</td> <td style="text-align:right;">4</td> <td style="text-align:right;">4 </td></tr> <tr> <td>23</td> <td style="text-align:right;">8</td> <td style="text-align:right;">8 </td></tr> <tr> <td>24</td> <td style="text-align:right;">10hex</td> <td style="text-align:right;">16dec </td></tr> <tr> <td>25</td> <td style="text-align:right;">20hex</td> <td style="text-align:right;">32dec </td></tr> <tr> <td>26</td> <td style="text-align:right;">40hex</td> <td style="text-align:right;">64dec </td></tr> <tr> <td>27</td> <td style="text-align:right;">80hex</td> <td style="text-align:right;">128dec </td></tr> <tr> <td>28</td> <td style="text-align:right;">100hex</td> <td style="text-align:right;">256dec </td></tr> <tr> <td>29</td> <td style="text-align:right;">200hex</td> <td style="text-align:right;">512dec </td></tr> <tr> <td>2A (210dec)</td> <td style="text-align:right;">400hex</td> <td style="text-align:right;">1,024dec </td></tr> <tr> <td>2B (211dec)</td> <td style="text-align:right;">800hex</td> <td style="text-align:right;">2,048dec </td></tr> <tr> <td>2C (212dec)</td> <td style="text-align:right;">1,000hex</td> <td style="text-align:right;">4,096dec </td></tr> <tr> <td>2D (213dec)</td> <td style="text-align:right;">2,000hex</td> <td style="text-align:right;">8,192dec </td></tr> <tr> <td>2E (214dec)</td> <td style="text-align:right;">4,000hex</td> <td style="text-align:right;">16,384dec </td></tr> <tr> <td>2F (215dec)</td> <td style="text-align:right;">8,000hex</td> <td style="text-align:right;">32,768dec </td></tr> <tr> <td>210 (216dec)</td> <td style="text-align:right;">10,000hex</td> <td style="text-align:right;">65,536dec </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Cultural_history">Cultural history</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=20" title="Edit section: Cultural history">edit</a>]</div> The traditional <a href="/wiki/Chinese_units_of_measurement" title="Chinese units of measurement">Chinese units of measurement</a> were base-16. For example, one jīn (斤) in the old system equals sixteen <a href="/wiki/Tael" title="Tael">taels</a>. The <a href="/wiki/Suanpan" title="Suanpan">suanpan</a> (Chinese <a href="/wiki/Abacus" title="Abacus">abacus</a>) can be used to perform hexadecimal calculations such as additions and subtractions.<a href="#cite_note-33">[33]</a> As with the <a href="/wiki/Duodecimal" title="Duodecimal">duodecimal</a> system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.<a href="#cite_note-34">[34]</a> Some proposals unify standard measures so that they are multiples of 16.<a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a> An early such proposal was put forward by <a href="/wiki/John_W._Nystrom" title="John W. Nystrom">John W. Nystrom</a> in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862.<a href="#cite_note-nystrom-37">[37]</a> Nystrom among other things suggested <a href="/wiki/Hexadecimal_time" title="Hexadecimal time">hexadecimal time</a>, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day.<a href="#cite_note-38">[38]</a> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/hexadecimal" class="extiw" title="wiktionary:hexadecimal">hexadecimal</a> in Wiktionary, the free dictionary.</div></div> </div> The word hexadecimal is first recorded in 1952.<a href="#cite_note-39">[39]</a> It is <a href="/wiki/Macaronic" class="mw-redirect" title="Macaronic">macaronic</a> in the sense that it combines <a href="/wiki/Greek_language" title="Greek language">Greek</a> ἕξ (hex) "six" with <a href="/wiki/Latin" title="Latin">Latinate</a> -decimal. The all-Latin alternative <a href="https://en.wiktionary.org/wiki/sexadecimal" class="extiw" title="wikt:sexadecimal">sexadecimal</a> (compare the word <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> for base 60) is older, and sees at least occasional use from the late 19th century.<a href="#cite_note-40">[40]</a> It is still in use in the 1950s in <a href="/wiki/Bendix_Corporation" title="Bendix Corporation">Bendix</a> documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex.<a href="#cite_note-41">[41]</a> Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.) Terminology and notation did not become settled until the end of the 1960s. In 1969, <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary, quaternary, etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively.<a href="#cite_note-42">[42]</a> Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".<a href="#cite_note-43">[43]</a><a href="#cite_note-44">[44]</a> The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the <a href="/wiki/Fortran_IV" class="mw-redirect" title="Fortran IV">Fortran IV</a> manual for <a href="/wiki/IBM_System/360" title="IBM System/360">IBM System/360</a>, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.<a href="#cite_note-45">[45]</a> As noted above, alternative notations were used by <a href="/wiki/NEC" title="NEC">NEC</a> (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">CACM</a> complains that <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!</blockquote> Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system".<a href="#cite_note-Martin_1968-19">[19]</a> He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as <a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi numerals</a>, and later in a <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>), and that the recent <a href="/wiki/ASCII" title="ASCII">ASCII</a> standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions. <div class="mw-heading mw-heading2"><h2 id="Base16_(transfer_encoding)">Base16 (transfer encoding)</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=21" title="Edit section: Base16 (transfer encoding)">edit</a>]</div> Base16 (as a proper name without a space) can also refer to a <a href="/wiki/Binary_to_text_encoding" class="mw-redirect" title="Binary to text encoding">binary to text encoding</a> belonging to the same family as <a href="/wiki/Base32" title="Base32">Base32</a>, <a href="/wiki/Base58" class="mw-redirect" title="Base58">Base58</a>, and <a href="/wiki/Base64" title="Base64">Base64</a>. In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using one of 16 symbols from the <a href="/wiki/ASCII" title="ASCII">ASCII</a> character set. Although any 16 symbols from the ASCII character set can be used, in practice, the ASCII digits "0"–"9" and the letters "A"–"F" (or the lowercase "a"–"f") are always chosen in order to align with standard written notation for hexadecimal numbers. There are several advantages of Base16 encoding: <ul><li>Most programming languages already have facilities to parse ASCII-encoded hexadecimal</li> <li>Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively</li> <li>The symbols 0–9 and A–F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table.</li> <li>Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "<a href="/wiki/Nibble" title="Nibble">nibble</a>"), making it more efficient in hardware than Base32 and Base64</li></ul> The main disadvantages of Base16 encoding are: <ul><li>Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.</li> <li>Possible added complexity of having to accept both uppercase and lowercase letters</li></ul> Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the <a href="/wiki/World_Wide_Web_Consortium" title="World Wide Web Consortium">W3C</a> standard for <a href="/wiki/Percent-encoding" title="Percent-encoding">URL percent encoding</a>, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=22" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Base32" title="Base32">Base32</a>, <a href="/wiki/Base64" title="Base64">Base64</a> (content encoding schemes)</li> <li><a href="/wiki/Hexadecimal_time" title="Hexadecimal time">Hexadecimal time</a></li> <li><a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">IBM hexadecimal floating-point</a></li> <li><a href="/wiki/Hex_editor" title="Hex editor">Hex editor</a></li> <li><a href="/wiki/Hex_dump" title="Hex dump">Hex dump</a></li> <li><a href="/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula" title="Bailey–Borwein–Plouffe formula">Bailey–Borwein–Plouffe formula</a> (BBP)</li> <li><a href="/wiki/Hexspeak" title="Hexspeak">Hexspeak</a></li> <li><a href="/wiki/P_notation" class="mw-redirect" title="P notation">P notation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Hexadecimal&action=edit&section=23" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.ionos.co.uk/digitalguide/server/know-how/hexadecimal-system/">"The hexadecimal system"</a>. Ionos Digital Guide. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220826084201/https://www.ionos.co.uk/digitalguide/server/know-how/hexadecimal-system/">Archived</a> from the original on 2022-08-26. Retrieved 2022-08-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ionos+Digital+Guide&rft.atitle=The+hexadecimal+system&rft_id=https%3A%2F%2Fwww.ionos.co.uk%2Fdigitalguide%2Fserver%2Fknow-how%2Fhexadecimal-system%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1986" class="citation book cs1">Knuth, Donald Ervin (1986). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/12973034">The TeXbook</a>. Duane Bibby. Reading, Mass. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-13447-0" title="Special:BookSources/0-201-13447-0"><bdi>0-201-13447-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/12973034">12973034</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220116012119/http://www.worldcat.org/oclc/12973034">Archived</a> from the original on 2022-01-16. Retrieved 2022-03-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+TeXbook&rft.place=Reading%2C+Mass.&rft.date=1986&rft_id=info%3Aoclcnum%2F12973034&rft.isbn=0-201-13447-0&rft.aulast=Knuth&rft.aufirst=Donald+Ervin&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F12973034&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> The string <code>"\x1B[0m\x1B[25;1H"</code> specifies the character sequence <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">Esc [ 0 m Esc [ 2 5; 1 H. These are the escape sequences used on an <a href="/wiki/ANSI_escape_code" title="ANSI escape code">ANSI terminal</a> that reset the character set and color, and then move the cursor to line 25. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2000.pdf">"The Unicode Standard, Version 7"</a> (PDF). Unicode. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303175510/http://www.unicode.org/charts/PDF/U2000.pdf">Archived</a> (PDF) from the original on 2016-03-03. Retrieved 2018-10-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Unicode&rft.atitle=The+Unicode+Standard%2C+Version+7&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU2000.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://modula2.org/reference/vocabulary.php">"Modula-2 – Vocabulary and representation"</a>. Modula −2. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151213053318/http://www.modula2.org/reference/vocabulary.php">Archived</a> from the original on 2015-12-13. Retrieved 2015-11-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Modula+%E2%88%922&rft.atitle=Modula-2+%E2%80%93+Vocabulary+and+representation&rft_id=http%3A%2F%2Fmodula2.org%2Freference%2Fvocabulary.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.fpgatutorial.com/vhdl-types-and-conversions#vhdl-assign-data">"An Introduction to VHDL Data Types"</a>. FPGA Tutorial. 2020-05-10. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200823094252/https://www.fpgatutorial.com/vhdl-types-and-conversions/#vhdl-assign-data">Archived</a> from the original on 2020-08-23. Retrieved 2020-08-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=FPGA+Tutorial&rft.atitle=An+Introduction+to+VHDL+Data+Types&rft.date=2020-05-10&rft_id=https%3A%2F%2Fwww.fpgatutorial.com%2Fvhdl-types-and-conversions%23vhdl-assign-data&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.lispworks.com/documentation/HyperSpec/Body/v_rd_bas.htm">"*read-base* variable in Common Lisp"</a>. CLHS. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160203221612/http://www.lispworks.com/documentation/HyperSpec/Body/v_rd_bas.htm">Archived</a> from the original on 2016-02-03. Retrieved 2015-01-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CLHS&rft.atitle=%2Aread-base%2A+variable+in+Common+Lisp&rft_id=http%3A%2F%2Fwww.lispworks.com%2Fdocumentation%2FHyperSpec%2FBody%2Fv_rd_bas.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.lispworks.com/documentation/HyperSpec/Body/v_pr_bas.htm#STprint-baseST">"*print-base* variable in Common Lisp"</a>. CLHS. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141226172420/http://www.lispworks.com/documentation/HyperSpec/Body/v_pr_bas.htm#STprint-baseST">Archived</a> from the original on 2014-12-26. Retrieved 2015-01-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CLHS&rft.atitle=%2Aprint-base%2A+variable+in+Common+Lisp&rft_id=http%3A%2F%2Fwww.lispworks.com%2Fdocumentation%2FHyperSpec%2FBody%2Fv_pr_bas.htm%23STprint-baseST&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a rel="nofollow" class="external text" href="http://www.atarimagazines.com/compute/issue56/107_1_MSX_IS_COMING.php">MSX is Coming — Part 2: Inside MSX</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101124111223/http://www.atarimagazines.com/compute/issue56/107_1_MSX_IS_COMING.php">Archived</a> 2010-11-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <a href="/wiki/Compute!" title="Compute!">Compute!</a>, issue 56, January 1985, p. 52 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> BBC BASIC programs are not fully portable to <a href="/wiki/Microsoft_BASIC" title="Microsoft BASIC">Microsoft BASIC</a> (without modification) since the latter takes <code>&</code> to prefix <a href="/wiki/Octal" title="Octal">octal</a> values. (Microsoft BASIC primarily uses <code>&O</code> to prefix octal, and it uses <code>&H</code> to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060422004336/http://www.web-colors-explained.com/hex.php">"Hexadecimal web colors explained"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.web-colors-explained.com/hex.php">the original</a> on 2006-04-22. 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Retrieved 2019-06-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.ic.unicamp.br&rft.atitle=ISO-8859-1+%28ISO+Latin+1%29+Character+Encoding&rft_id=https%3A%2F%2Fwww.ic.unicamp.br%2F~stolfi%2FEXPORT%2Fwww%2FISO-8859-1-Encoding.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-Savard_2018_CA-13">^ <a href="#cite_ref-Savard_2018_CA_13-0">a</a> <a href="#cite_ref-Savard_2018_CA_13-1">b</a> <a href="#cite_ref-Savard_2018_CA_13-2">c</a> <a href="#cite_ref-Savard_2018_CA_13-3">d</a> <a href="#cite_ref-Savard_2018_CA_13-4">e</a> <a href="#cite_ref-Savard_2018_CA_13-5">f</a> <a href="#cite_ref-Savard_2018_CA_13-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSavard2018" class="citation web cs1">Savard, John J. G. (2018) [2005]. <a rel="nofollow" class="external text" href="http://www.quadibloc.com/comp/cp02.htm">"Computer Arithmetic"</a>. quadibloc. The Early Days of Hexadecimal. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180716102439/http://www.quadibloc.com/comp/cp02.htm">Archived</a> from the original on 2018-07-16. Retrieved 2018-07-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=quadibloc&rft.atitle=Computer+Arithmetic&rft.pages=The+Early+Days+of+Hexadecimal&rft.date=2018&rft.aulast=Savard&rft.aufirst=John+J.+G.&rft_id=http%3A%2F%2Fwww.quadibloc.com%2Fcomp%2Fcp02.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-Bendix-14"><a href="#cite_ref-Bendix_14-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"2.1.3 Sexadecimal notation". <a rel="nofollow" class="external text" href="http://bitsavers.trailing-edge.com/pdf/bendix/g-15/G15D_Programmers_Ref_Man.pdf">G15D Programmer's Reference Manual</a> (PDF). Los Angeles, CA, US: <a href="/wiki/Bendix_Computer" class="mw-redirect" title="Bendix Computer">Bendix Computer</a>, Division of <a href="/wiki/Bendix_Aviation_Corporation" class="mw-redirect" title="Bendix Aviation Corporation">Bendix Aviation Corporation</a>. p. 4. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170601222212/http://bitsavers.trailing-edge.com/pdf/bendix/g-15/G15D_Programmers_Ref_Man.pdf">Archived</a> (PDF) from the original on 2017-06-01. Retrieved 2017-06-01. <q>This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations, we arrive at a notation called sexadecimal (usually "hex" in conversation because nobody wants to abbreviate "sex"). The symbols in the sexadecimal language are the ten decimal digits and on the G-15 typewriter, the letters "u", "v", "w", "x", "y", and "z". These are arbitrary markings; other computers may use different alphabet characters for these last six digits.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.1.3+Sexadecimal+notation&rft.btitle=G15D+Programmer%27s+Reference+Manual&rft.place=Los+Angeles%2C+CA%2C+US&rft.pages=4&rft.pub=Bendix+Computer%2C+Division+of+Bendix+Aviation+Corporation&rft_id=http%3A%2F%2Fbitsavers.trailing-edge.com%2Fpdf%2Fbendix%2Fg-15%2FG15D_Programmers_Ref_Man.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-Illiac-I-15"><a href="#cite_ref-Illiac-I_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillNeagherMullerNash1956" class="citation web cs1">Gill, S.; Neagher, R. E.; Muller, D. E.; Nash, J. P.; Robertson, J. E.; Shapin, T.; Whesler, D. J. (1956-09-01). Nash, J. P. (ed.). <a rel="nofollow" class="external text" href="http://www.textfiles.com/bitsavers/pdf/illiac/ILLIAC/ILLIAC_programming_Sep56.pdf">"ILLIAC Programming – A Guide to the Preparation of Problems For Solution by the University of Illinois Digital Computer"</a> (PDF). bitsavers.org (Fourth printing. Revised and corrected ed.). Urbana, Illinois, US: Digital Computer Laboratory, Graduate College, <a href="/wiki/University_of_Illinois" class="mw-redirect" title="University of Illinois">University of Illinois</a>. pp. 3–2. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170531153804/http://www.textfiles.com/bitsavers/pdf/illiac/ILLIAC/ILLIAC_programming_Sep56.pdf">Archived</a> (PDF) from the original on 2017-05-31. Retrieved 2014-12-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=bitsavers.org&rft.atitle=ILLIAC+Programming+%E2%80%93+A+Guide+to+the+Preparation+of+Problems+For+Solution+by+the+University+of+Illinois+Digital+Computer&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E2&rft.date=1956-09-01&rft.aulast=Gill&rft.aufirst=S.&rft.au=Neagher%2C+R.+E.&rft.au=Muller%2C+D.+E.&rft.au=Nash%2C+J.+P.&rft.au=Robertson%2C+J.+E.&rft.au=Shapin%2C+T.&rft.au=Whesler%2C+D.+J.&rft_id=http%3A%2F%2Fwww.textfiles.com%2Fbitsavers%2Fpdf%2Filliac%2FILLIAC%2FILLIAC_programming_Sep56.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-RP_1957_LGP-30-16"><a href="#cite_ref-RP_1957_LGP-30_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://ed-thelen.org/comp-hist/lgp-30-man.html#R4.13">Royal Precision Electronic Computer LGP – 30 Programming Manual</a>. Port Chester, New York: <a href="/wiki/Royal_McBee_Corporation" class="mw-redirect" title="Royal McBee Corporation">Royal McBee Corporation</a>. April 1957. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170531153004/http://ed-thelen.org/comp-hist/lgp-30-man.html">Archived</a> from the original on 2017-05-31. Retrieved 2017-05-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Royal+Precision+Electronic+Computer+LGP+%E2%80%93+30+Programming+Manual&rft.place=Port+Chester%2C+New+York&rft.pub=Royal+McBee+Corporation&rft.date=1957-04&rft_id=http%3A%2F%2Fed-thelen.org%2Fcomp-hist%2Flgp-30-man.html%23R4.13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the <a href="/wiki/LGP-30" title="LGP-30">LGP-30</a>'s 6-bit character code.) </li> <li id="cite_note-PERM-17"><a href="#cite_ref-PERM_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMantheyLeibrandt2002" class="citation web cs1 cs1-prop-foreign-lang-source">Manthey, Steffen; Leibrandt, Klaus (2002-07-02). <a rel="nofollow" class="external text" href="http://www.manthey.cc/sites/seminars/src/History.pdf">"Die PERM und ALGOL"</a> (PDF) (in German). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181003172841/http://www.manthey.cc/sites/seminars/src/History.pdf">Archived</a> (PDF) from the original on 2018-10-03. Retrieved 2018-05-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Die+PERM+und+ALGOL&rft.date=2002-07-02&rft.aulast=Manthey&rft.aufirst=Steffen&rft.au=Leibrandt%2C+Klaus&rft_id=http%3A%2F%2Fwww.manthey.cc%2Fsites%2Fseminars%2Fsrc%2FHistory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-NEC_1960_NEAC-1103-18"><a href="#cite_ref-NEC_1960_NEAC-1103_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://archive.computerhistory.org/resources/text/NEC/NEC.1103.1958102646285.pdf">NEC Parametron Digital Computer Type NEAC-1103</a> (PDF). Tokyo, Japan: <a href="/wiki/Nippon_Electric_Company_Ltd." class="mw-redirect" title="Nippon Electric Company Ltd.">Nippon Electric Company Ltd.</a> 1960. Cat. 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The Open Group Base Specifications (Issue 7, IEEE Std 1003.1, 2013 ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160621211105/http://pubs.opengroup.org/onlinepubs/9699919799/functions/printf.html">Archived</a> from the original on 2016-06-21. Retrieved 2016-06-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Open+Group+Base+Specifications&rft.atitle=dprintf%2C+fprintf%2C+printf%2C+snprintf%2C+sprintf+%E2%80%93+print+formatted+output&rft.date=2013&rft.au=The+IEEE+and+The+Open+Group&rft_id=http%3A%2F%2Fpubs.opengroup.org%2Fonlinepubs%2F9699919799%2Ffunctions%2Fprintf.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-Mano-Ciletti-31">^ <a href="#cite_ref-Mano-Ciletti_31-0">a</a> <a href="#cite_ref-Mano-Ciletti_31-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManoCiletti2013" class="citation book cs1">Mano, M. Morris; Ciletti, Michael D. (2013). Digital Design – With an Introduction to the Verilog HDL (Fifth ed.). <a href="/wiki/Pearson_Education" title="Pearson Education">Pearson Education</a>. pp. 6, 8–10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-277420-8" title="Special:BookSources/978-0-13-277420-8"><bdi>978-0-13-277420-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Design+%E2%80%93+With+an+Introduction+to+the+Verilog+HDL&rft.pages=6%2C+%3Cspan+class%3D%22nowrap%22%3E8-%3C%2Fspan%3E10&rft.edition=Fifth&rft.pub=Pearson+Education&rft.date=2013&rft.isbn=978-0-13-277420-8&rft.aulast=Mano&rft.aufirst=M.+Morris&rft.au=Ciletti%2C+Michael+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Inaccurate, up to the 21st digit out of 46 in decimal </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://totton.idirect.com/soroban/Hex_as/">"算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus"</a>. totton.idirect.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190706221609/http://totton.idirect.com/soroban/Hex_as/">Archived</a> from the original on 2019-07-06. Retrieved 2019-06-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=totton.idirect.com&rft.atitle=%E7%AE%97%E7%9B%A4+Hexadecimal+Addition+%26+Subtraction+on+a+Chinese+Abacus&rft_id=http%3A%2F%2Ftotton.idirect.com%2Fsoroban%2FHex_as%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.hauptmech.com/base42">"Base 4^2 Hexadecimal Symbol Proposal"</a>. Hauptmech. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211020192525/http://www.hauptmech.com/base42/wiki/index.php?title=Main_Page">Archived</a> from the original on 2021-10-20. Retrieved 2008-09-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Hauptmech&rft.atitle=Base+4%5E2+Hexadecimal+Symbol+Proposal&rft_id=http%3A%2F%2Fwww.hauptmech.com%2Fbase42&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.intuitor.com/hex/">"Intuitor Hex Headquarters"</a>. Intuitor. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100904144850/http://www.intuitor.com/hex/">Archived</a> from the original on 2010-09-04. Retrieved 2018-10-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Intuitor&rft.atitle=Intuitor+Hex+Headquarters&rft_id=http%3A%2F%2Fwww.intuitor.com%2Fhex%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiemietz2003" class="citation web cs1">Niemietz, Ricardo Cancho (2003-10-21). <a rel="nofollow" class="external text" href="https://www.unicode.org/wg2/docs/n2677.pdf">"A proposal for addition of the six Hexadecimal digits (A-F) to Unicode"</a> (PDF). ISO/IEC JTC1/SC2/WG2. Retrieved 2024-06-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+proposal+for+addition+of+the+six+Hexadecimal+digits+%28A-F%29+to+Unicode&rft.pub=ISO%2FIEC+JTC1%2FSC2%2FWG2&rft.date=2003-10-21&rft.aulast=Niemietz&rft.aufirst=Ricardo+Cancho&rft_id=https%3A%2F%2Fwww.unicode.org%2Fwg2%2Fdocs%2Fn2677.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-nystrom-37"><a href="#cite_ref-nystrom_37-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNystrom1862" class="citation book cs1">Nystrom, John William (1862). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_aNYGAAAAYAAJ">Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base</a>. Philadelphia: Lippincott.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Project+of+a+New+System+of+Arithmetic%2C+Weight%2C+Measure+and+Coins%3A+Proposed+to+be+called+the+Tonal+System%2C+with+Sixteen+to+the+Base&rft.place=Philadelphia&rft.pub=Lippincott&rft.date=1862&rft.aulast=Nystrom&rft.aufirst=John+William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_aNYGAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> Nystrom (1862), p. 33: "In expressing time, angle of a circle, or points on the compass, the unit tim should be noted as integer, and parts thereof as tonal fractions, as 5·86 tims is five times and metonby [*"sutim and metonby" John Nystrom accidentally gives part of the number in decimal names; in Nystrom's pronunciation scheme, 5=su, 8=me, 6=by, c.f. <a rel="nofollow" class="external text" href="http://www.unifoundry.com/tonal/index.html">unifoundry.com</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210519080658/http://www.unifoundry.com/tonal/index.html">Archived</a> 2021-05-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> ]." </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> C. E. Fröberg, Hexadecimal Conversion Tables, Lund (1952). </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> The Century Dictionary of 1895 has sexadecimal in the more general sense of "relating to sixteen". An early explicit use of sexadecimal in the sense of "using base 16" is found also in 1895, in the Journal of the American Geographical Society of New York, vols. 27–28, p. 197. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartzman1994" class="citation book cs1">Schwartzman, Steven (1994). The Words of Mathematics: An etymological dictionary of mathematical terms used in English. The Mathematical Association of America. p. 105. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-511-9" title="Special:BookSources/0-88385-511-9"><bdi>0-88385-511-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Words+of+Mathematics%3A+An+etymological+dictionary+of+mathematical+terms+used+in+English&rft.pages=105&rft.pub=The+Mathematical+Association+of+America&rft.date=1994&rft.isbn=0-88385-511-9&rft.aulast=Schwartzman&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHexadecimal" class="Z3988"> s.v. hexadecimal </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Knuth, Donald. (1969). <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, Volume 2. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-03802-1" title="Special:BookSources/0-201-03802-1">0-201-03802-1</a>. (Chapter 17.) </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> 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, 15 September 1859. See pages and 33 and 41. </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> 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/20160624070056/https://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296">Archived</a> 2016-06-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Philadelphia, 1887; pages 296–366. See pages 317 and 322. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a rel="nofollow" class="external text" href="http://www.bitsavers.org/pdf/ibm/360/fortran/C28-6515-6_FORTRAN_IV_Language_1966.pdf">IBM System/360 FORTRAN IV Language</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210519073220/http://www.bitsavers.org/pdf/ibm/360/fortran/C28-6515-6_FORTRAN_IV_Language_1966.pdf">Archived</a> 2021-05-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (1966), p. 13. </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Hexadecimal&oldid=1276934418">https://en.wikipedia.org/w/index.php?title=Hexadecimal&oldid=1276934418</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:Hexadecimal_numeral_system" title="Category:Hexadecimal numeral system">Hexadecimal numeral system</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><li><a href="/wiki/Category:Positional_numeral_systems" title="Category:Positional numeral systems">Positional 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:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><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:CS1_location_test" title="Category:CS1 location test">CS1 location test</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_March_2020" title="Category:Use dmy dates from March 2020">Use dmy dates from March 2020</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 21 February 2025, at 16:57 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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