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href="#Natural_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Natural languages</span> </div> </a> <ul id="toc-Natural_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Base_36_as_senary_compression" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Base_36_as_senary_compression"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Base 36 as senary compression</span> </div> </a> <ul id="toc-Base_36_as_senary_compression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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Available in 27 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-27" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Lio%CC%8Dk-ch%C3%ACn-hoat" title="Lio̍k-chìn-hoat – Minnan" lang="nan" hreflang="nan" data-title="Lio̍k-chìn-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C5%A0estkov%C3%A1_soustava" title="Šestková soustava – Czech" lang="cs" hreflang="cs" data-title="Šestková soustava" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Sen%C3%A4r#Senäres_Zahlensystem" title="Senär – German" lang="de" hreflang="de" data-title="Senär" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%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="Εξαδικό σύστημα αρίθμησης – Greek" lang="el" hreflang="el" data-title="Εξαδικό σύστημα αρίθμησης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_senario" title="Sistema senario – Spanish" lang="es" hreflang="es" data-title="Sistema senario" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sesuma_nombrosistemo" title="Sesuma nombrosistemo – Esperanto" lang="eo" hreflang="eo" data-title="Sesuma nombrosistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki-sistema_seitar" title="Zenbaki-sistema seitar – Basque" lang="eu" hreflang="eu" data-title="Zenbaki-sistema seitar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%BE%D8%A7%DB%8C%D9%87_%DB%B3%DB%B6" title="دستگاه اعداد پایه ۳۶ – Persian" lang="fa" hreflang="fa" data-title="دستگاه اعداد پایه ۳۶" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_s%C3%A9naire" title="Système sénaire – French" lang="fr" hreflang="fr" data-title="Système sénaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A1%EC%A7%84%EB%B2%95" title="육진법 – Korean" lang="ko" hreflang="ko" data-title="육진법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_numerico_senario" title="Sistema numerico senario – Italian" lang="it" hreflang="it" data-title="Sistema numerico senario" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_sen%C3%A8" title="Sistèm senè – Haitian Creole" lang="ht" hreflang="ht" data-title="Sistèm senè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hatos_sz%C3%A1mrendszer" title="Hatos számrendszer – Hungarian" lang="hu" hreflang="hu" data-title="Hatos számrendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%85%AD%E9%80%B2%E6%B3%95" title="六進法 – Japanese" lang="ja" hreflang="ja" data-title="六進法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Oltilik_sanoq_sistemasi" title="Oltilik sanoq sistemasi – Uzbek" lang="uz" hreflang="uz" data-title="Oltilik sanoq sistemasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sz%C3%B3stkowy_system_liczbowy" title="Szóstkowy system liczbowy – Polish" lang="pl" hreflang="pl" data-title="Szóstkowy system liczbowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_senar" title="Sistem senar – Romanian" lang="ro" hreflang="ro" data-title="Sistem senar" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Senary" title="Senary – Simple English" lang="en-simple" hreflang="en-simple" data-title="Senary" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Senary" title="Senary – Somali" lang="so" hreflang="so" data-title="Senary" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Senaarij%C3%A4rjestelm%C3%A4" title="Senaarijärjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Senaarijärjestelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sen%C3%A4ra_talsystemet" title="Senära talsystemet – Swedish" lang="sv" hreflang="sv" data-title="Senära talsystemet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A5%E0%B8%82%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%AB%E0%B8%81" title="เลขฐานหก – Thai" lang="th" hreflang="th" data-title="เลขฐานหก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A8%D1%96%D1%81%D1%82%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Шісткова система числення – Ukrainian" lang="uk" hreflang="uk" data-title="Шісткова система числення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_l%E1%BB%A5c_ph%C3%A2n" title="Hệ lục phân – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ lục phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%85%AD%E9%80%B2%E5%88%B6" title="六進制 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="六進制" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%AD%E9%80%B2%E5%88%B6" title="六進制 – Cantonese" lang="yue" hreflang="yue" data-title="六進制" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic</a></li> <li><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul> <hr /> <ul><li><a href="/wiki/Bengali_numerals" title="Bengali numerals">Bengali</a></li> <li><a href="/wiki/Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li> <li><a href="/wiki/Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li> <li><a href="/wiki/Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li> <li><a href="/wiki/Odia_numerals" title="Odia numerals">Odia</a></li> <li><a href="/wiki/Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li> <li><a href="/wiki/Tamil_numerals" title="Tamil numerals">Tamil</a></li> <li><a href="/wiki/Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li> <li><a href="/wiki/Telugu_script#Numerals" title="Telugu script">Telugu</a></li> <li><a href="/wiki/Kannada_script#Numerals" title="Kannada script">Kannada</a></li> <li><a href="/wiki/Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul> <hr /> <ul><li><a href="/wiki/Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li> <li><a href="/wiki/Balinese_numerals" title="Balinese numerals">Balinese</a></li> <li><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></li> <li><a href="/wiki/Javanese_numerals" title="Javanese numerals">Javanese</a></li> <li><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></li> <li><a href="/wiki/Lao_script#Numerals" title="Lao script">Lao</a></li> <li><a href="/wiki/Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li> <li><a href="/wiki/Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li> <li><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a> <ul><li><a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li> <li><a href="/wiki/Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li> <li><a href="/wiki/Japanese_numerals" title="Japanese numerals">Japanese</a></li> <li><a href="/wiki/Korean_numerals" title="Korean numerals">Korean</a></li> <li><a href="/wiki/Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul> <hr /> <dl><dt>Historic</dt></dl> <ul><li><a href="/wiki/Counting_rods" title="Counting rods">Counting rods</a></li> <li><a href="/wiki/Tangut_numerals" title="Tangut numerals">Tangut</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul> <hr /> <dl><dt><a href="/wiki/Ancient_history" title="Ancient history">Ancient</a></dt></dl> <ul><li><a href="/wiki/Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul> <hr /> <dl><dt><a href="/wiki/Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl> <ul><li><a href="/wiki/Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li> <li><a href="/wiki/Maya_numerals" title="Maya numerals">Mayan</a></li> <li><a href="/wiki/Muisca_numerals" title="Muisca numerals">Muisca</a></li> <li><a href="/wiki/Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li> <li><a href="/wiki/Quipu" title="Quipu">Quipu</a></li> <li><a href="/wiki/Rumi_Numeral_Symbols" title="Rumi Numeral Symbols">Rumi</a></li></ul> <hr /> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li> <li><a href="/wiki/Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</a> (Iñupiaq)</li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="/wiki/Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Common radices/bases</dt></dl> <ul><li><a href="/wiki/Binary_number" title="Binary number">2</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">3</a></li> <li><a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">4</a></li> <li><a href="/wiki/Quinary" title="Quinary">5</a></li> <li><a class="mw-selflink selflink">6</a></li> <li><a href="/wiki/Octal" title="Octal">8</a></li> <li><a href="/wiki/Decimal" title="Decimal">10</a></li> <li><a href="/wiki/Duodecimal" title="Duodecimal">12</a></li> <li><a href="/wiki/Hexadecimal" title="Hexadecimal">16</a></li> <li><a href="/wiki/Vigesimal" title="Vigesimal">20</a></li> <li><a href="/wiki/Sexagesimal" title="Sexagesimal">60</a></li></ul> <hr /> <dl><dt><a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl> <ul><li><a href="/wiki/Bijective_numeration" title="Bijective numeration">Bijective</a><span class="nowrap"> </span>(<a href="/wiki/Unary_numeral_system" title="Unary numeral system">1</a>)</li> <li><a href="/wiki/Signed-digit_representation" title="Signed-digit representation">Signed-digit</a><span class="nowrap"> </span>(<a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li> <li><a href="/wiki/Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<a href="/wiki/Factorial_number_system" title="Factorial number system">factorial</a>)</li> <li><a href="/wiki/Negative_base" title="Negative base">Negative</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex</a><span class="nowrap"> </span>(<a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li> <li><a href="/wiki/Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<a href="/wiki/Golden_ratio_base" title="Golden ratio base">φ</a>)</li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></li></ul></div></div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Non-alphabetic</dt></dl> <ul><li><a href="/wiki/Aegean_numerals" title="Aegean numerals">Aegean</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic</a></li> <li><a href="/wiki/Aztec_script#Numerals" title="Aztec script">Aztec</a></li> <li><a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li> <li><a href="/wiki/Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li> <li><a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li> <li><a href="/wiki/Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li> <li><a href="/wiki/Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li> <li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric counting</a></li> <li><a href="/wiki/Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li> <li><a href="/wiki/Roman_numerals" title="Roman numerals">Roman</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li></ul> <hr /> <dl><dt><a href="/wiki/Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl> <ul><li><a href="/wiki/Abjad_numerals" title="Abjad numerals">Abjad</a></li> <li><a href="/wiki/Armenian_numerals" title="Armenian numerals">Armenian</a></li> <li><a href="/wiki/Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a> <ul><li><a href="/wiki/Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li> <li><a href="/wiki/Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li> <li><a href="/wiki/Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li> <li><a href="/wiki/Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li> <li><a href="/wiki/Georgian_numerals" title="Georgian numerals">Georgian</a></li> <li><a href="/wiki/Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek</a></li> <li><a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numeral_systems" title="Template:Numeral systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numeral_systems" title="Template talk:Numeral systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numeral_systems" title="Special:EditPage/Template:Numeral systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>senary</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'s' in 'sigh'">s</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'n' in 'nigh'">n</span><span title="/ər/: 'er' in 'letter'">ər</span><span title="/i/: 'y' in 'happy'">i</span></span>,<span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'s' in 'sigh'">s</span><span title="/ɛ/: 'e' in 'dress'">ɛ</span><span title="'n' in 'nigh'">n</span><span title="/ər/: 'er' in 'letter'">ər</span><span title="/i/: 'y' in 'happy'">i</span></span>/</a></span></span>) <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a> (also known as <b>base-6</b>, <b>heximal</b>, or <b>seximal</b>) has <a href="/wiki/6" title="6">six</a> as its <a href="/wiki/Radix" title="Radix">base</a>. It has been adopted independently by a small number of cultures. Like the <a href="/wiki/Decimal" title="Decimal">decimal</a> base 10, the base is a <a href="/wiki/Semiprime" title="Semiprime">semiprime</a>, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a <a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">superior highly composite number</a>, many of the arguments made in favor of the <a href="/wiki/Duodecimal" title="Duodecimal">duodecimal</a> system also apply to the senary system. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of digits in the senary system is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{6}=\lbrace 0,1,2,3,4,5\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{6}=\lbrace 0,1,2,3,4,5\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1612b3c85ae54bab238e30f3dde047efcee08481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.414ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{6}=\lbrace 0,1,2,3,4,5\rbrace }"></span>, with the <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linear order</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<1<2<3<4<5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mn>1</mn> <mo><</mo> <mn>2</mn> <mo><</mo> <mn>3</mn> <mo><</mo> <mn>4</mn> <mo><</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<1<2<3<4<5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef67a2c6519883a47864e2cba0e57c136b74959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.467ex; height:2.176ex;" alt="{\displaystyle 0<1<2<3<4<5}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{6}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{6}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87920a6f93a9e7389944b62eb0894b87c3018c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.846ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{6}^{*}}"></span> be the <a href="/wiki/Kleene_closure" class="mw-redirect" title="Kleene closure">Kleene closure</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d5248d53b99a899fb92de8f56f99f4f1227b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{6}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"></span> is the operation of <a href="/wiki/String_concatenation" class="mw-redirect" title="String concatenation">string concatenation</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in {\mathcal {D}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in {\mathcal {D}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9479c8a4b1757bd103681592586bf01781ba19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.948ex; height:2.676ex;" alt="{\displaystyle a,b\in {\mathcal {D}}^{*}}"></span>. The senary number system for <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa9b13decff68385951bef6561f5b8ae3c8d72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{6}}"></span> is the <a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}_{6}^{*}/\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{6}^{*}/\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753022e8d0ad3e018dd9642d43a41a81fb08ffa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{6}^{*}/\sim }"></span> equipped with a <a href="/wiki/Shortlex_order" title="Shortlex order">shortlex order</a>, where the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace n\in {\mathcal {D}}_{6}^{*},n\sim 0n\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mi>n</mi> <mo>∼</mo> <mn>0</mn> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace n\in {\mathcal {D}}_{6}^{*},n\sim 0n\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f371fad792c17a17051318ea8403fbf2dbd835fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.491ex; height:2.843ex;" alt="{\displaystyle \lbrace n\in {\mathcal {D}}_{6}^{*},n\sim 0n\rbrace }"></span>. As <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa9b13decff68385951bef6561f5b8ae3c8d72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{6}}"></span> has a shortlex order, it is <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to the natural numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=2" title="Edit section: Mathematical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright" style="text-align:center"> <caption>Senary <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a> </caption> <tbody><tr> <th>×</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>10</th> <th> </th></tr> <tr> <th>1 </th> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>10</td> <td> </td></tr> <tr> <th>2 </th> <td>2</td> <td>4</td> <td>10</td> <td>12</td> <td>14</td> <td>20</td> <td> </td></tr> <tr> <th>3 </th> <td>3</td> <td>10</td> <td>13</td> <td>20</td> <td>23</td> <td>30</td> <td> </td></tr> <tr> <th>4 </th> <td>4</td> <td>12</td> <td>20</td> <td>24</td> <td>32</td> <td>40</td> <td> </td></tr> <tr> <th>5 </th> <td>5</td> <td>14</td> <td>23</td> <td>32</td> <td>41</td> <td>50</td> <td> </td></tr> <tr> <th>10 </th> <td>10</td> <td>20</td> <td>30</td> <td>40</td> <td>50</td> <td>100</td> <td> </td></tr></tbody></table> <p>When expressed in senary, all <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> other than 2 and 3 have 1 or 5 as the final digit. In senary, the prime numbers are written: </p> <dl><dd>2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, ... (sequence <span class="nowrap external"><a href="//oeis.org/A004680" class="extiw" title="oeis:A004680">A004680</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>That is, for every prime number <i>p</i> greater than 3, one has the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> relations that either <i>p</i> ≡ 1 or 5 (mod 6) (that is, 6 divides either <i>p</i> − 1 or <i>p</i> − 5); the final digit is a 1 or a 5. This is proved by contradiction. </p><p>For any integer <i>n</i>: </p> <ul><li>If <i>n</i> ≡ 0 (mod 6), 6 | <i>n</i></li> <li>If <i>n</i> ≡ 2 (mod 6), 2 | <i>n</i></li> <li>If <i>n</i> ≡ 3 (mod 6), 3 | <i>n</i></li> <li>If <i>n</i> ≡ 4 (mod 6), 2 | <i>n</i></li></ul> <p>Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple <a href="/wiki/Divisibility_test" class="mw-redirect" title="Divisibility test">divisibility tests</a> for many numbers. </p><p>Furthermore, all even <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a> besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form <span class="nowrap">2<sup><i>p</i> – 1</sup>(2<sup><i>p</i></sup> – 1)</span>, where <span class="nowrap">2<sup><i>p</i></sup> − 1</span> is prime. </p><p>Senary is also the largest number base <i>r</i> that has no <a href="/wiki/Totative" title="Totative">totatives</a> other than 1 and <i>r</i> − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do. </p><p>If a number is divisible by 2, then the final digit of that number, when expressed in senary, is 0, 2, or 4. If a number is divisible by 3, then the final digit of that number in senary is 0 or 3. A number is divisible by 4 if its penultimate digit is odd and its final digit is 2, or its penultimate digit is even and its final digit is 0 or 4. A number is divisible by 5 if the sum of its senary digits is divisible by 5 (the equivalent of <a href="/wiki/Casting_out_nines" title="Casting out nines">casting out nines</a> in decimal). If a number is divisible by 6, then the final digit of that number is 0. To determine whether a number is divisible by 7, one can sum its alternate digits and subtract those sums; if the result is divisible by 7, the number is divisible by 7, similar to the "11" divisibility test in decimal. </p> <div class="mw-heading mw-heading2"><h2 id="Fractions">Fractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=3" title="Edit section: Fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because six is the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of the first two <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> and is adjacent to the next two prime numbers, many senary fractions have simple representations: </p> <table class="wikitable"> <tbody><tr> <td colspan="3" align="center"><b>Decimal base</b><br /><small>Prime factors of the base: <span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span></small><br /><small>Prime factors of one below the base: <span style="color:Blue"><b>3</b></span></small><br /><small>Prime factors of one above the base: <span style="color:Magenta"><b>11</b></span></small><br /> </td> <td colspan="3" align="center"><b>Senary base</b><br /><small>Prime factors of the base: <span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span></small><br /><small>Prime factors of one below the base: <span style="color:Blue"><b>5</b></span></small><br /><small>Prime factors of one above the base: <span style="color:Magenta"><b>7 (=11<sub>6</sub>)</b></span></small><br /> </td></tr> <tr> <th align="center">Fraction </th> <th align="center"><small>Prime factors<br />of the denominator</small> </th> <th align="center">Positional representation </th> <th align="center">Positional representation </th> <th align="center"><small>Prime factors<br />of the denominator</small> </th> <th align="center">Fraction </th></tr> <tr> <td align="center"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.5</b> </td> <td><b>0.3</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.</b>3333... = <b>0.</b><span style="text-decoration:overline;">3</span> </td> <td><b>0.2</b> </td> <td align="center"><span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.25</b> </td> <td><b>0.13</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>5</b></span> </td> <td><b>0.2</b> </td> <td bgcolor="#d0d0d0"><b>0.</b>1111... = <b>0.</b><span style="text-decoration:overline;">1</span> </td> <td align="center"><span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.1</b><span style="text-decoration:overline;">6</span> </td> <td><b>0.1</b> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>7</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">142857</span> </td> <td bgcolor="#d0d0d0"><b>0.</b><span style="text-decoration:overline;">05</span> </td> <td align="center"><span style="color:Magenta"><b>11</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">11</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.125</b> </td> <td><b>0.043</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.</b><span style="text-decoration:overline;">1</span> </td> <td><b>0.04</b> </td> <td align="center"><span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">13</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span> </td> <td><b>0.1</b> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">3</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">14</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">11</span></span>⁠</span> </td> <td align="center"><span style="color:Magenta"><b>11</b></span> </td> <td bgcolor="#d0d0d0"><b>0.</b><span style="text-decoration:overline;">09</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">0313452421</span> </td> <td align="center"><span style="color:Red"><b>15</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">15</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.08</b><span style="text-decoration:overline;">3</span> </td> <td><b>0.03</b> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">20</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">13</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>13</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">076923</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">024340531215</span> </td> <td align="center"><span style="color:Red"><b>21</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">21</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">14</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">714285</span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">23</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">22</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">15</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span>, <span style="color:Green"><b>5</b></span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">6</span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">2</span> </td> <td align="center"><span style="color:Green"><b>3</b></span>, <span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">23</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">16</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.0625</b> </td> <td><b>0.0213</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">24</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">17</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>17</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">0588235294117647</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">0204122453514331</span> </td> <td align="center"><span style="color:Red"><b>25</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">25</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">18</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">5</span> </td> <td><b>0.02</b> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">30</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">19</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>19</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">052631578947368421</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">015211325</span> </td> <td align="center"><span style="color:Red"><b>31</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">31</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">20</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>5</b></span> </td> <td><b>0.05</b> </td> <td bgcolor="#d0d0d0"><b>0.01</b><span style="text-decoration:overline;">4</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">32</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">21</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">047619</span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">14</span> </td> <td align="center"><span style="color:Green"><b>3</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">33</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">22</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">45</span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">1345242103</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>15</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">34</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">23</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>23</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">0434782608695652173913</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">01322030441</span> </td> <td align="center"><span style="color:Red"><b>35</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">35</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">24</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.041</b><span style="text-decoration:overline;">6</span> </td> <td><b>0.013</b> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">40</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">25</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>5</b></span> </td> <td><b>0.04</b> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">01235</span> </td> <td align="center"><span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">41</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">26</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>13</b></span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">384615</span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">121502434053</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>21</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">42</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">27</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">037</span> </td> <td><b>0.012</b> </td> <td align="center"><span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">43</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">28</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#808080"><b>0.03</b><span style="text-decoration:overline;">571428</span> </td> <td bgcolor="#d0d0d0"><b>0.01</b><span style="text-decoration:overline;">14</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">44</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">29</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>29</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">0344827586206896551724137931</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">01124045443151</span> </td> <td align="center"><span style="color:Red"><b>45</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">45</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">30</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span>, <span style="color:Green"><b>5</b></span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">3</span> </td> <td bgcolor="#d0d0d0"><b>0.0</b><span style="text-decoration:overline;">1</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span>, <span style="color:Blue"><b>5</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">50</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">31</span></span>⁠</span> </td> <td align="center"><span style="color:Red"><b>31</b></span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">032258064516129</span> </td> <td bgcolor="#808080"><b>0.</b><span style="text-decoration:overline;">010545</span> </td> <td align="center"><span style="color:Red"><b>51</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">51</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">32</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td><b>0.03125</b> </td> <td><b>0.01043</b> </td> <td align="center"><span style="color:Green"><b>2</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">52</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">33</span></span>⁠</span> </td> <td align="center"><span style="color:Blue"><b>3</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td bgcolor="#d0d0d0"><b>0.</b><span style="text-decoration:overline;">03</span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">1031345242</span> </td> <td align="center"><span style="color:Green"><b>3</b></span>, <span style="color:Red"><b>15</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">53</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">34</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>17</b></span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">2941176470588235</span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">1020412245351433</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Red"><b>25</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">54</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">35</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>5</b></span>, <span style="color:Red"><b>7</b></span> </td> <td bgcolor="#808080"><b>0.0</b><span style="text-decoration:overline;">285714</span> </td> <td bgcolor="#d0d0d0"><b>0.</b><span style="text-decoration:overline;">01</span> </td> <td align="center"><span style="color:Blue"><b>5</b></span>, <span style="color:Magenta"><b>11</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">55</span></span>⁠</span> </td></tr> <tr> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">36</span></span>⁠</span> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Blue"><b>3</b></span> </td> <td bgcolor="#d0d0d0"><b>0.02</b><span style="text-decoration:overline;">7</span> </td> <td><b>0.01</b> </td> <td align="center"><span style="color:Green"><b>2</b></span>, <span style="color:Green"><b>3</b></span> </td> <td align="center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">100</span></span>⁠</span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Finger_counting">Finger counting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=4" title="Edit section: Finger counting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finger_counting" class="mw-redirect" title="Finger counting">Finger counting</a></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:248px;max-width:248px"><div class="trow"><div class="tsingle" style="width:122px;max-width:122px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Chinesische.Zahl.Drei.jpg" class="mw-file-description"><img alt="3" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Chinesische.Zahl.Drei.jpg/120px-Chinesische.Zahl.Drei.jpg" decoding="async" width="120" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4b/Chinesische.Zahl.Drei.jpg 1.5x" data-file-width="144" data-file-height="150" /></a></span></div></div><div class="tsingle" style="width:122px;max-width:122px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Chinesische.Zahl.Vier.jpg" class="mw-file-description"><img alt="4" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Chinesische.Zahl.Vier.jpg/120px-Chinesische.Zahl.Vier.jpg" decoding="async" width="120" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5b/Chinesische.Zahl.Vier.jpg 1.5x" data-file-width="149" data-file-height="150" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">34<sub>senary</sub> = 22<sub>decimal</sub>, in senary finger counting</div></div></div></div> <p>Each regular human hand may be said to have six unambiguous positions; a fist, one finger extended, two, three, four, and then all five fingers extended. </p><p>If the right hand is used to represent a unit (0 to 5), and the left to represent the multiples of 6, then it becomes possible for one person to represent the values from zero to 55<sub>senary</sub> (35<sub>decimal</sub>) with their fingers, rather than the usual ten obtained in standard finger counting. e.g. if three fingers are extended on the left hand and four on the right, 34<sub>senary</sub> is represented. This is equivalent to <b>3</b> × 6 + <b>4</b>, which is 22<sub>decimal</sub>. </p><p>Additionally, this method is the least abstract way to count using two hands that reflects the concept of <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a>, as the movement from one position to the next is done by switching from one hand to another. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with <a href="/wiki/Chinese_number_gestures" title="Chinese number gestures">Chinese number gestures</a>. As senary finger counting also deviates only beyond 5, this counting method rivals the simplicity of traditional counting methods, a fact which may have implications for the teaching of positional notation to young students. </p><p>Which hand is used for the 'sixes' and which the units is down to preference on the part of the counter; however, when viewed from the counter's perspective, using the left hand as the most significant digit correlates with the written representation of the same senary number. Flipping the 'sixes' hand around to its backside may help to further disambiguate which hand represents the 'sixes' and which represents the units. The downside to senary counting, however, is that without prior agreement two parties would be unable to utilize this system, being unsure which hand represents sixes and which hand represents ones, whereas decimal-based counting (with numbers beyond 5 being expressed by an open palm and additional fingers) being essentially a <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary</a> system only requires the other party to count the number of extended fingers. </p><p>In <a href="/wiki/NCAA_basketball" class="mw-redirect" title="NCAA basketball">NCAA basketball</a>, the players' <a href="/wiki/Number_(sports)" title="Number (sports)">uniform numbers</a> are restricted to be senary numbers of at most two digits, so that the referees can signal which player committed an infraction by using this finger-counting system.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>More abstract <a href="/wiki/Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a> systems, such as <a href="/wiki/Chisanbop" title="Chisanbop">chisanbop</a> or <a href="/wiki/Finger_binary" title="Finger binary">finger binary</a>, allow counting to 99, 1023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historian <a href="/wiki/Bede" title="Bede">Bede</a>, described in the first chapter of his work <i>De temporum ratione</i>, (725), titled "<i>Tractatus de computo, vel loquela per gestum digitorum</i>," a system which allowed counting up to 9,999 on two hands.<sup id="cite_ref-handsums_2-0" class="reference"><a href="#cite_note-handsums-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-laputan_3-0" class="reference"><a href="#cite_note-laputan-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Natural_languages">Natural languages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=5" title="Edit section: Natural languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed.<sup id="cite_ref-plankS_5-0" class="reference"><a href="#cite_note-plankS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Ndom_language" title="Ndom language">Ndom language</a> of <a href="/wiki/Indonesian_New_Guinea" class="mw-redirect" title="Indonesian New Guinea">Indonesian New Guinea</a> is reported to have senary numerals.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <i>Mer</i> means 6, <i>mer an thef</i> means 6 × 2 = 12, <i>nif</i> means 36, and <i>nif thef</i> means 36 × 2 = 72. </p><p>Another example from <a href="/wiki/Papua_New_Guinea" title="Papua New Guinea">Papua New Guinea</a> are the <a href="/wiki/Yam_languages" title="Yam languages">Yam languages</a>. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up to 6<sup>6</sup> for some of the languages. One example is <a href="/wiki/K%C3%B3mnzo_language" title="Kómnzo language">Komnzo</a> with the following numerals: <i>nibo</i> (6<sup>1</sup>), <i>fta</i> (6<sup>2</sup> [36]), <i>taruba</i> (6<sup>3</sup> [216]), <i>damno</i> (6<sup>4</sup> [1296]), <i>wärämäkä</i> (6<sup>5</sup> [7776]), <i>wi</i> (6<sup>6</sup> [46656]). </p><p>Some <a href="/wiki/Niger-Congo_languages" class="mw-redirect" title="Niger-Congo languages">Niger-Congo languages</a> have been reported to use a senary number system, usually in addition to another, such as <a href="/wiki/Decimal" title="Decimal">decimal</a> or <a href="/wiki/Vigesimal" title="Vigesimal">vigesimal</a>.<sup id="cite_ref-plankS_5-1" class="reference"><a href="#cite_note-plankS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Proto-Uralic_language" title="Proto-Uralic language">Proto-Uralic</a> has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 and 9) subtractively from ten suggests that this may not be so.<sup id="cite_ref-plankS_5-2" class="reference"><a href="#cite_note-plankS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Base_36_as_senary_compression">Base 36 as senary compression</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=6" title="Edit section: Base 36 as senary compression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Base 36" redirects here. For the encoding scheme used to represent binary data as text, see <a href="/wiki/Base36" title="Base36">Base36</a>.</div> <p>For some purposes, senary might be too small a base for convenience. This can be worked around by using its square, base 36 (hexatrigesimal), as then conversion is facilitated by simply making the following replacements: </p> <table class="wikitable"> <tbody><tr align="right"> <th>Decimal </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> <td>12</td> <td>13</td> <td>14</td> <td>15</td> <td>16</td> <td>17 </td></tr> <tr align="right"> <th>Base 6 </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>10</td> <td>11</td> <td>12</td> <td>13</td> <td>14</td> <td>15</td> <td>20</td> <td>21</td> <td>22</td> <td>23</td> <td>24</td> <td>25 </td></tr> <tr align="right"> <th>Base 36 </th> <td><code>0</code></td> <td><code>1</code></td> <td><code>2</code></td> <td><code>3</code></td> <td><code>4</code></td> <td><code>5</code></td> <td><code>6</code></td> <td><code>7</code></td> <td><code>8</code></td> <td><code>9</code></td> <td><code>A</code></td> <td><code>B</code></td> <td><code>C</code></td> <td><code>D</code></td> <td><code>E</code></td> <td><code>F</code></td> <td><code>G</code></td> <td><code>H</code> </td></tr> <tr> <td style="font: 0.5em/0.5em serif;" colspan="19">  </td></tr> <tr align="right"> <th>Decimal </th> <td>18</td> <td>19</td> <td>20</td> <td>21</td> <td>22</td> <td>23</td> <td>24</td> <td>25</td> <td>26</td> <td>27</td> <td>28</td> <td>29</td> <td>30</td> <td>31</td> <td>32</td> <td>33</td> <td>34</td> <td>35 </td></tr> <tr align="right"> <th>Base 6 </th> <td>30</td> <td>31</td> <td>32</td> <td>33</td> <td>34</td> <td>35</td> <td>40</td> <td>41</td> <td>42</td> <td>43</td> <td>44</td> <td>45</td> <td>50</td> <td>51</td> <td>52</td> <td>53</td> <td>54</td> <td>55 </td></tr> <tr align="right"> <th>Base 36 </th> <td><code>I</code></td> <td><code>J</code></td> <td><code>K</code></td> <td><code>L</code></td> <td><code>M</code></td> <td><code>N</code></td> <td><code>O</code></td> <td><code>P</code></td> <td><code>Q</code></td> <td><code>R</code></td> <td><code>S</code></td> <td><code>T</code></td> <td><code>U</code></td> <td><code>V</code></td> <td><code>W</code></td> <td><code>X</code></td> <td><code>Y</code></td> <td><code>Z</code> </td></tr></tbody></table> <p>Thus, the base-36 number 3ARK<sub>36</sub> is equal to the senary number 3144332<sub>6</sub>. In decimal, it is 153,920. </p><p>The choice of 36 as a <a href="/wiki/Radix" title="Radix">radix</a> is convenient in that the digits can be represented using the <a href="/wiki/Hindu%E2%80%93Arabic_numerals" class="mw-redirect" title="Hindu–Arabic numerals">Arabic numerals</a> 0–9 and the <a href="/wiki/Latin_alphabet" title="Latin alphabet">Latin letters</a> A–Z; this choice is the basis of the <a href="/wiki/Base36" title="Base36">base36</a> encoding scheme. The compression effect of 36 being the square of 6 causes a lot of patterns and representations to be shorter in base 36: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">9<sub>10</sub></span></span>⁠</span> = 0.04<sub>6</sub> = 0.4<sub>36</sub></dd></dl> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">16<sub>10</sub></span></span>⁠</span> = 0.0213<sub>6</sub> = 0.29<sub>36</sub></dd></dl> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5<sub>10</sub></span></span>⁠</span> = 0.<span style="text-decoration:overline;">1</span><sub>6</sub> = 0.<span style="text-decoration:overline;">7</span><sub>36</sub></dd></dl> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7<sub>10</sub></span></span>⁠</span> = 0.<span style="text-decoration:overline;">05</span><sub>6</sub> = 0.<span style="text-decoration:overline;">5</span><sub>36</sub></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Diceware" title="Diceware">Diceware</a> method to encode base-6 values into pronounceable passwords.</li> <li><a href="/wiki/Base36" title="Base36">Base36</a> encoding scheme</li> <li><a href="/wiki/ADFGVX_cipher" title="ADFGVX cipher">ADFGVX cipher</a> to encrypt text into a series of effectively senary digits</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free 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"Origins of Northern Costanoan ʃak:en 'six':A Reconsideration of Senary Counting in Utian". <i>International Journal of American Linguistics</i>. <b>71</b> (1): 87–101. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F430579">10.1086/430579</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.1086/430579">10.1086/430579</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144384806">144384806</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+American+Linguistics&rft.atitle=Origins+of+Northern+Costanoan+%CA%83ak%3Aen+%27six%27%3AA+Reconsideration+of+Senary+Counting+in+Utian&rft.volume=71&rft.issue=1&rft.pages=87-101&rft.date=2018-05-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144384806%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F10.1086%2F430579%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F430579&rft.aulast=Blevins&rft.aufirst=Juliette&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASenary" class="Z3988"></span></span> </li> <li id="cite_note-plankS-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-plankS_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-plankS_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-plankS_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlank2009" class="citation journal cs1">Plank, Frans (26 April 2009). <a rel="nofollow" class="external text" href="https://ling.sprachwiss.uni-konstanz.de/pages/home/plank/for_download/publications/151_Plank_SenerySummary_2009.pdf">"Senary summary so far"</a> <span class="cs1-format">(PDF)</span>. <i>Linguistic Typology</i>. <b>13</b> (2). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2FLITY.2009.016">10.1515/LITY.2009.016</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55100862">55100862</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160406105208/http://ling.uni-konstanz.de/pages/home/plank/for_download/publications/151_Plank_SenerySummary_2009.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2016-04-06<span class="reference-accessdate">. Retrieved <span class="nowrap">August 31,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linguistic+Typology&rft.atitle=Senary+summary+so+far&rft.volume=13&rft.issue=2&rft.date=2009-04-26&rft_id=info%3Adoi%2F10.1515%2FLITY.2009.016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55100862%23id-name%3DS2CID&rft.aulast=Plank&rft.aufirst=Frans&rft_id=https%3A%2F%2Fling.sprachwiss.uni-konstanz.de%2Fpages%2Fhome%2Fplank%2Ffor_download%2Fpublications%2F151_Plank_SenerySummary_2009.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASenary" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOwens2001" class="citation journal cs1">Owens, Kay (April 2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF03217098">"The work of Glendon Lean on the counting systems of Papua New Guinea and Oceania"</a></span>. <i><a href="/wiki/Mathematics_Education_Research_Journal" title="Mathematics Education Research Journal">Mathematics Education Research Journal</a></i>. <b>13</b> (1): 47–71. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001MEdRJ..13...47O">2001MEdRJ..13...47O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03217098">10.1007/BF03217098</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1033-2170">1033-2170</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:161535519">161535519</a><span class="reference-accessdate">. Retrieved <span class="nowrap">August 31,</span> 2022</span> – via Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Education+Research+Journal&rft.atitle=The+work+of+Glendon+Lean+on+the+counting+systems+of+Papua+New+Guinea+and+Oceania&rft.volume=13&rft.issue=1&rft.pages=47-71&rft.date=2001-04&rft_id=info%3Adoi%2F10.1007%2FBF03217098&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A161535519%23id-name%3DS2CID&rft.issn=1033-2170&rft_id=info%3Abibcode%2F2001MEdRJ..13...47O&rft.aulast=Owens&rft.aufirst=Kay&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF03217098&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASenary" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOwens2001" class="citation cs2">Owens, Kay (2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150926003303/http://www.uog.ac.pg/glec/Key/Kay/owens131.htm">"The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania"</a>, <i>Mathematics Education Research Journal</i>, <b>13</b> (1): 47–71, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001MEdRJ..13...47O">2001MEdRJ..13...47O</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03217098">10.1007/BF03217098</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:161535519">161535519</a>, archived from <a rel="nofollow" class="external text" href="http://www.uog.ac.pg/glec/Key/Kay/owens131.htm">the original</a> on 2015-09-26</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Education+Research+Journal&rft.atitle=The+Work+of+Glendon+Lean+on+the+Counting+Systems+of+Papua+New+Guinea+and+Oceania&rft.volume=13&rft.issue=1&rft.pages=47-71&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A161535519%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF03217098&rft_id=info%3Abibcode%2F2001MEdRJ..13...47O&rft.aulast=Owens&rft.aufirst=Kay&rft_id=http%3A%2F%2Fwww.uog.ac.pg%2Fglec%2FKey%2FKay%2Fowens131.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASenary" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Senary&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.seximal.net/">Comprehensive base six resource</a></li> <li><a rel="nofollow" class="external text" href="http://shacktoms.org/base-six/base-six.htm">Shack's base six dialect</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathsisfun.com/numbers/convert-base.php?to=senary">Senary base conversion</a></li> <li><a rel="nofollow" class="external text" href="https://sooeet.com/math/base-converter.php">Number-Base Radix Converter (Sooeet)</a></li> <li><a rel="nofollow" class="external text" href="https://sezimal.tauga.online/calculator">Calculator</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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