View source for List of numeral systems

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{{About||different types of numbers, such as rational numbers, real numbers, complex numbers, etc|List of types of numbers}}
{{use mdy dates|date=May 2023}}
{{Numeral systems}}
{{Contains special characters
| special    = uncommon [[Unicode]] characters
| fix        = Help:Multilingual support
| error      = [[Specials (Unicode block)#Replacement character|question marks, boxes, or other symbols]]
| characters = the intended characters
| image      = Replacement character.svg
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}}

There are many different [[numeral system]]s, that is, [[writing system]]s for expressing [[numbers]].

==By culture / time period==

"A ''base'' is a natural number B whose ''powers'' (B multiplied by itself some number of times) are specially designated within a numerical system."<ref name="Chrisomalis2004">{{cite journal |last1=Chrisomalis |first1=Stephen |date=2004 |title=A cognitive typology for numerical notation |journal=Cambridge Archaeological Journal |volume=14 |issue=1 |pages=37–52 |doi=10.1017/S0959774304000034 }}</ref>{{rp|38}} The term is not equivalent to [[radix]], as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers.<ref name="Chrisomalis2004"/> Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base).
{| class="wikitable sortable"
|-
! Name 
! data-sort-type=number | Base 
! Sample 
! data-sort-type=number | Approx. First Appearance
|-
|[[Proto-cuneiform numerals]] || 10{{resize|88%|&}}60 ||  || {{sort|-3500|c. 3500–2000 BCE}}
|-
|[[Indus script|Indus numerals]]|| unknown{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200 pp. 330-333]}}||  || {{sort|-3501|c. 3500–1900 BCE}}{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200 pp. 330-333]}}
|-
|[[Proto-Elamite#Proto-Elamite script|Proto-Elamite numerals]]|| 10{{resize|88%|&}}60 ||  || {{sort|-3101|3100 BCE}}
|-
|[[Sumerian numerals]]|| 10{{resize|88%|&}}60 ||  || {{sort|-3100|3100 BCE}}
|-
| [[Egyptian numerals]] || 10 || <hiero size=8>Z1 V20 V1 M12 D50 I8 I7 C11</hiero> || {{sort|-3000|3000 BCE}}
|-
|[[Babylonian numerals]]|| 10{{resize|88%|&}}60 || [[File:Babylonian 1.svg|15px]] [[File:Babylonian 2.svg|15px]] [[File:Babylonian 3.svg|15px]] [[File:Babylonian 4.svg|15px]] [[File:Babylonian 5.svg|15px]] [[File:Babylonian 6.svg|15px]] [[File:Babylonian 7.svg|15px]] [[File:Babylonian 8.svg|15px]] [[File:Babylonian 9.svg|15px]] [[File:Babylonian 10.svg|15px]] || {{sort|-2000|2000 BCE}}
|-
| [[Aegean numerals]] || 10 || 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  &nbsp;( [[File:Aegean numeral 1.svg|10px|frameless|1]] [[File:Aegean numeral 2.svg|10px|frameless|2]] [[File:Aegean numeral 3.svg|10px|frameless|3]] [[File:Aegean numeral 4.svg|10px|frameless|4]] [[File:Aegean numeral 5.svg|10px|frameless|5]] [[File:Aegean numeral 6.svg|10px|frameless|6]] [[File:Aegean numeral 7.svg|10px|frameless|7]] [[File:Aegean numeral 8.svg|10px|frameless|8]] [[File:Aegean numeral 9.svg|10px|frameless|9]] )<br /> 𐄐 𐄑 𐄒 𐄓  𐄔 𐄕 𐄖 𐄗 𐄘 &nbsp;( [[File:Aegean numeral 10.svg|10px|frameless|10]] [[File:Aegean numeral 20.svg|10px|frameless|20]] [[File:Aegean numeral 30.svg|10px|frameless|30]] [[File:Aegean numeral 40.svg|10px|frameless|40]] [[File:Aegean numeral 50.svg|10px|frameless|50]] [[File:Aegean numeral 60.svg|10px|frameless|60]] [[File:Aegean numeral 70.svg|10px|frameless|70]] [[File:Aegean numeral 80.svg|10px|frameless|80]] [[File:Aegean numeral 90.svg|10px|frameless|90]] )<br />  𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 &nbsp;( [[File:Aegean numeral 100.svg|10px|frameless|100]] [[File:Aegean numeral 200.svg|10px|frameless|200]] [[File:Aegean numeral 300.svg|10px|frameless|300]] [[File:Aegean numeral 400.svg|10px|frameless|400]] [[File:Aegean numeral 500.svg|10px|frameless|500]] [[File:Aegean numeral 600.svg|10px|frameless|600]] [[File:Aegean numeral 700.svg|10px|frameless|700]] [[File:Aegean numeral 800.svg|10px|frameless|800]] [[File:Aegean numeral 900.svg|10px|frameless|900]] )<br /> 𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪 &nbsp;( [[File:Aegean numeral 1000.svg|10px|frameless|1000]] [[File:Aegean numeral 2000.svg|10px|frameless|2000]] [[File:Aegean numeral 3000.svg|10px|frameless|3000]] [[File:Aegean numeral 4000.svg|10px|frameless|4000]] [[File:Aegean numeral 5000.svg|10px|frameless|5000]] [[File:Aegean numeral 6000.svg|10px|frameless|6000]] [[File:Aegean numeral 7000.svg|10px|frameless|7000]] [[File:Aegean numeral 8000.svg|10px|frameless|8000]] [[File:Aegean numeral 9000.svg|10px|frameless|9000]] )<br /> 𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 &nbsp;( [[File:Aegean numeral 10000.svg|10px|frameless|10000]] [[File:Aegean numeral 20000.svg|10px|frameless|20000]] [[File:Aegean numeral 30000.svg|10px|frameless|30000]] [[File:Aegean numeral 40000.svg|10px|frameless|40000]] [[File:Aegean numeral 50000.svg|10px|frameless|50000]] [[File:Aegean numeral 60000.svg|10px|frameless|60000]] [[File:Aegean numeral 70000.svg|10px|frameless|70000]] [[File:Aegean numeral 80000.svg|10px|frameless|80000]] [[File:Aegean numeral 90000.svg|10px|frameless|90000]] )|| {{sort|-1500|1500 BCE}}
|-
|[[Chinese numerals]]<br />[[Japanese numerals]]<br />[[Korean numerals]] ([[Sino-Korean vocabulary|Sino-Korean]])<br />[[Vietnamese numerals]] ([[Sino-Vietnamese vocabulary|Sino-Vietnamese]])|| 10 ||
零一二三四五六七八九十百千萬億 (Default, [[Traditional Chinese]])<br />
〇一二三四五六七八九十百千万亿 (Default, [[Simplified Chinese]])
| {{sort|-1300|1300 BCE}}
|-
| [[Roman numerals]] ||  5{{resize|88%|&}}10|| I V X L C D M || {{sort|-1000|1000 BCE}}<ref name="Chrisomalis2004"/>
|-
| [[Hebrew numerals]] ||  10|| {{tt|א ב ג ד ה ו ז ח ט<br />י כ ל מ נ ס ע פ צ<br />ק ר ש ת ך ם ן ף ץ}}  || {{sort|-800|800 BCE}}
|-
| [[Indian numerals]] || 10 ||

[[Bengali numerals|Bengali]] ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯

[[Devanagari numerals|Devanagari]] ० १ २ ३ ४ ५ ६ ७ ८ ९

[[Gujarati numerals|Gujarati]] ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯

[[Kannada script#Numerals|Kannada]] ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯

[[Malayalam numerals|Malayalam]] ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯

[[Odia numerals|Odia]] ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯

[[Punjabi language|Punjabi]] ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯

[[Tamil numerals|Tamil]] ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

[[Telugu script#Numerals|Telugu]] ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯

[[Tibetan numerals|Tibetan]] <big>༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩</big>

[[Eastern Arabic numerals|Urdu]] <big>۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹</big> 
| {{sort|-750|750–500 BCE}}
|-
| [[Greek numerals]] || 10 || ō α β γ δ ε ϝ ζ η θ ι <br /> ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ || {{sort|-400|<400 BCE}}
|-
|[[Kharosthi|Kharosthi numerals]]
|4{{resize|88%|&}}10
|𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀
|{{sort|-401|<400–250 BCE}}<ref>{{cite web |last1=Glass |first1=Andrew |last2=Baums |first2=Stefan |last3=Salomon |first3=Richard |date=2003-09-18 |title=Proposal to Encode Kharoṣ ṭhī in Plane 1 of ISO/IEC 10646 |url=https://www.unicode.org/L2/L2003/03314-kharoshthi.pdf |website=[[Unicode|Unicode.org]]}}</ref>
|-
| [[Phoenician alphabet#Numerals|Phoenician numerals]] || 10 || 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖  <ref>{{cite web |last1=Everson|first1=Michael |title=Proposal to add two numbers for the Phoenician script |url=https://www.unicode.org/L2/L2007/07206-n3284-phoenician.pdf |website=UTC Document Register |publisher=Unicode Consortium |at=L2/07-206 (WG2 N3284)|date=2007-07-25}}</ref> ||  {{sort|-250|<250 BCE}}<ref name=Cajori>{{cite book|last1=Cajori|first1=Florian|author-link1=Florian Cajori|title=A History Of Mathematical Notations Vol I|date=Sep 1928|publisher=The Open Court Company|page=18|url=https://archive.org/stream/historyofmathema031756mbp#page/n37/mode/2up|access-date=5 June 2017|language=en}}</ref>
|-
| Chinese [[rod numerals]] || 10 || 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 || {{sort|1|1st Century}}
|-
| [[Coptic alphabet#Numerals|Coptic numerals]] || 10 || Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ || {{sort|100|2nd Century}}
|-
| [[Ge'ez numerals]] || 10 || ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱  <br /> ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ <br /> ፻ <br /> ፼ <ref>{{cite web |title=Ethiopic (Unicode block)|url=https://unicode.org/charts/PDF/U1200.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|200|3rd–4th Century}}<br />15th Century (Modern Style)<ref name="Chrisomalis2010">{{Cite book |url=https://books.google.com/books?id=ux--OWgWvBQC&pg=PA135 |title = Numerical Notation: A Comparative History |language=en |publisher=[[Cambridge University Press]] |isbn=978-0-521-87818-0 |last1=Chrisomalis| first1=Stephen |date=2010 }}</ref>{{rp|135-136}}
|-
| [[Armenian numerals]] || 10 || Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ || {{sort|400|Early 5th Century}}
|-
| [[Khmer numerals]] || 10 || ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ || {{sort|600|Early 7th Century}}
|-
| [[Thai numerals]] || 10 || ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ || {{sort|601|7th Century}}{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200 p. 200]}}
|-
| [[Abjad numerals]] || 10 || غ ظ ض ذ خ ث ت ش ر ق ص ف ع س ن م ل ك ي ط ح ز و هـ د ج ب ا|| {{sort|680|<8th Century}}
|-
| [[Chinese numerals]] (financial) || 10 || 零壹貳參肆伍陸柒捌玖拾佰仟萬億 (T. Chinese)<br />零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (S. Chinese) || {{sort|690|late 7th/early 8th Century}}<ref>{{Cite web |last=Guo |first=Xianghe |date=2009-07-27 |title=武则天为反贪发明汉语大写数字——中新网 |trans-title=Wu Zetian invented Chinese capital numbers to fight corruption |url=https://www.chinanews.com.cn/hb/news/2009/07-27/1792519.shtml |access-date=2024-08-15 |website=中新社 [China News Service]}}</ref>
|-
| [[Eastern Arabic numerals]] || 10 ||٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ || {{sort|701|8th Century}}
|-
| [[Vietnamese numerals]] ([[Chữ Nôm]]) || 10 || 𠬠 𠄩 	𠀧 𦊚 𠄼 𦒹 	𦉱 𠔭 𠃩 || {{sort|799|<9th Century}}
|-
| [[Western Arabic numerals]] || 10 || 0 1 2 3 4 5 6 7 8 9 || {{sort|801|9th Century}}
|-
| [[Glagolitic numerals]] || 10 || Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... || {{sort|800|9th Century}}
|-
| [[Cyrillic numerals]] || 10 || а в г д е ѕ з и ѳ і ... || {{sort|900|10th Century}}
|-
| [[Rumi Numeral Symbols|Rumi numerals]] || 10 || [[File:Rumi numerals 1-9.svg|left|150px]]|| {{sort|900|10th Century}}
|-
| [[Burmese numerals]] || 10 || ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ || {{sort|1000|11th Century}}<ref>{{cite web|title=Burmese/Myanmar script and pronunciation|url=http://www.omniglot.com/writing/burmese.htm|website=Omniglot|access-date=5 June 2017}}</ref>
|-
| [[Tangut numerals]] || 10 || {{Tangut|𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗|lew ny so lyr ngwy chhiw sha ar gy gha}} || {{sort|1036|11th Century (1036)}}
|-
| [[Cistercian numerals]] || 10 || [[File:Cistercian numerals.svg|frameless|upright]] || {{sort|1200|13th Century}}
|-
| [[Maya numerals]] || 5{{resize|88%|&}}20 || [[Image:0 maia.svg|15px]] [[Image:1 maia.svg|15px]] [[Image:2 maia.svg|15px]] [[Image:3 maia.svg|15px]] [[Image:4 maia.svg|15px]] [[Image:5 maia.svg|15px]] [[Image:6 maia.svg|15px]] [[Image:7 maia.svg|15px]] [[Image:8 maia.svg|15px]] [[Image:9 maia.svg|15px]] [[Image:10 maia.svg|15px]] [[Image:11 maia.svg|15px]] [[Image:12 maia.svg|15px]] [[Image:13 maia.svg|15px]] [[Image:14 maia.svg|15px]] [[Image:15 maia.svg|15px]] [[Image:16 maia.svg|15px]] [[Image:17 maia.svg|15px]] [[Image:18 maia.svg|15px]] [[Image:19 maia.svg|15px]] || {{sort|1400|<15th Century}}
|-
| [[Muisca numerals]] || 20 || [[File:Muisca cyphers acc acosta humboldt zerda.svg|frameless|upright=1.5]] || {{sort|1399|<15th Century}}
|-
| [[Korean numerals]] ([[Hangul]]) || 10 || 영 일 이 삼 사 오 육 칠 팔 구 || {{sort|1443|15th Century (1443)}}
|-
| [[Aztec numerals]] || 20 || || {{sort|1500|16th Century}}
|-
| [[Sinhala numerals]] || 10 || |෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣<br />𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯  𑇰 𑇱 𑇲 𑇳 𑇴 || {{sort|1699|<18th Century}}
|-
| [[Pentadic numerals|Pentadic runes]] || 10 || [[File:Pentimal Runes 1 through 10.svg|frameless|upright]] || {{sort|1800|19th Century}}
|-
| [[Cherokee syllabary#Numerals|Cherokee numerals]] || 10 || [[File:Cherokee Numbers – cropped (1-20).png|frameless|upright=2]] || {{sort|1820|19th Century (1820s)}}
|-
| [[Vai syllabary#Digits|Vai numerals]] || 10 || ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ <ref>{{cite web |title=Vai (Unicode block)|url=https://unicode.org/charts/PDF/UA500.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1832|19th Century (1832)<ref name="Open Science Framework">{{cite web |last1=Kelly|first1=Piers|title=The invention, transmission and evolution of writing: Insights from the new scripts of West Africa|url=https://osf.io/preprints/socarxiv/253vc/download | website=Open Science Framework}}</ref>}}
|-
| [[Bamum script#Numbers|Bamum numerals]] || 10 || ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ <ref>{{cite web |title=Bamum (Unicode block)|url=https://unicode.org/charts/PDF/UA6A0.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1896|19th Century (1896)<ref name="Open Science Framework"/>}}
|-
| [[Mende Kikakui script|Mende Kikakui numerals]] || 10 || 𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 <ref>{{cite web |title=Mende Kikakui (Unicode block)|url=https://unicode.org/charts/PDF/U1E800.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1896|20th Century (1917)<ref>{{cite web |last1=Everson|first1=Michael|title=Proposal for encoding the Mende script in the SMP of the UCS|url=https://www.unicode.org/L2/L2011/11301r-n4133-mende.pdf | website=UTC Document Register |publisher=Unicode Consortium |at=L2/11-301R (WG2 N4133R)|date=2011-10-21}}</ref>}}
|-
| [[Osmanya script|Osmanya numerals]] || 10 || 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 || {{sort|1921|20th Century (1920s)}}
|-
| [[Medefaidrin#Numbers and dates|Medefaidrin numerals]] || 20 || 𖺀 𖺁/𖺔 𖺂/𖺕 𖺃/𖺖 𖺄 𖺅 𖺆 𖺇 𖺈 𖺉 𖺊 𖺋 𖺌 𖺍 𖺎 𖺏 𖺐 𖺑 𖺒 𖺓 <ref>{{cite web |title=Medefaidrin (Unicode block)|url=https://unicode.org/charts/PDF/U16E40.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1930|20th Century (1930s)<ref>{{cite web |last1=Rovenchak|first1=Andrij|title=Preliminary proposal for encoding the Medefaidrin (Oberi Okaime) script in the SMP of the UCS (Revised)|url=https://www.unicode.org/L2/L2015/15117r2-medefaidrin.pdf | website=UTC Document Register |publisher=Unicode Consortium |at=L2/L2015|date=2015-07-17}}</ref>}}
|-
| [[N'Ko script#Numerals|N'Ko numerals]] || 10 || ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ <ref>{{cite web |title=NKo (Unicode block)|url=https://unicode.org/charts/PDF/U07C0.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1949|20th Century (1949)<ref>{{cite web |last1=Donaldson|first1=Coleman|title=Clear Language: Script, Register And The N'ko Movement Of Manding-Speaking West Africa|url=https://www.unicode.org/L2/L2015/15117r2-medefaidrin.pdf | website=repository.upenn.edu |publisher=UPenn |date=2017-01-01}}</ref>}}
|-
| [[Pahawh Hmong#Numeral system|Hmong numerals]] || 10 || {{script|Hmng|𖭐}} {{script|Hmng|𖭑}} {{script|Hmng|𖭒}} {{script|Hmng|𖭓}} {{script|Hmng|𖭔}} {{script|Hmng|𖭕}} {{script|Hmng|𖭖}} {{script|Hmng|𖭗}} {{script|Hmng|𖭘}} {{script|Hmng|𖭑𖭐}} || {{sort|1959|20th Century (1959)}}
|-
| [[Garay alphabet#Numbers|Garay numerals]] || 10 || [[File:Garay numbers.png|Garay numbers]]<ref>{{cite web |title=Consideration of the encoding of Garay with updated user feedback (revised)|url=https://www.unicode.org/L2/L2022/22048-garay-script.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1961|20th Century (1961)<ref>{{cite web |last1=Everson|first1=Michael|title=Proposal for encoding the Garay script in the SMP of the UCS|url=https://www.unicode.org/L2/L2016/16069-n4709-garay-revision.pdf |website=UTC Document Register |publisher=Unicode Consortium |at=L2/L16-069 (WG2 N4709)|date=2016-03-22}}</ref>}}
|-
| [[Adlam script#Digits|Adlam numerals]]|| 10 || 𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 <ref>{{cite web |title=Adlam (Unicode block) |url=https://unicode.org/charts/PDF/U1E900.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1989|20th Century (1989)<ref>{{cite web |last1=Everson|first1=Michael|title=Revised proposal for encoding the Adlam script in the SMP of the UCS |url=https://www.unicode.org/L2/L2014/14219r-n4628-adlam.pdf | website=UTC Document Register |publisher=Unicode Consortium |at=L2/L14-219R (WG2 N4628R)|date=2014-10-28}}</ref>}}
|-
| [[Kaktovik numerals]] || 5{{resize|88%|&}}20 || {{Kaktovik digit|0}} {{Kaktovik digit|1}} {{Kaktovik digit|2}} {{Kaktovik digit|3}} {{Kaktovik digit|4}} {{Kaktovik digit|5}} {{Kaktovik digit|6}} {{Kaktovik digit|7}} {{Kaktovik digit|8}} {{Kaktovik digit|9}} {{Kaktovik digit|10}} {{Kaktovik digit|11}} {{Kaktovik digit|12}} {{Kaktovik digit|13}} {{Kaktovik digit|14}} {{Kaktovik digit|15}} {{Kaktovik digit|16}} {{Kaktovik digit|17}} {{Kaktovik digit|18}} {{Kaktovik digit|19}} <br /> 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 <ref>{{cite web |title=Kaktovik Numerals (Unicode block) |url=https://unicode.org/charts/PDF/U1D2C0.pdf | website=Unicode Character Code Charts|publisher=Unicode Consortium}}</ref>|| {{sort|1994|20th Century (1994)<ref>{{cite web |last1=Silvia|first1=Eduardo|title=Exploratory proposal to encode the Kaktovik numerals |url=https://www.unicode.org/L2/L2020/20070-kaktovik-numerals.pdf | website=UTC Document Register |publisher=Unicode Consortium |at=L2/20-070|date=2020-02-09}}</ref>}}
|-
|[[Sundanese numerals]]
|10
|᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹
|20th Century (1996)<ref>{{cite web |date=2008 |title=Direktori Aksara Sunda untuk Unicode |url=http://file.upi.edu/Direktori/FPBS/JUR._PEND._BHS._DAN_SASTRA_INDONESIA/197006242006041-TEDI_PERMADI/Direktori_Aksara_Sunda_untuk_Unicode.pdf |publisher=Pemerintah Provinsi Jawa Barat |language=id}}{{page needed|date=August 2024}}</ref>
|}<!-- Sundanese is not a typo of Sudanese  -->

==By type of notation==
{{anchor|Positional numeral systems}}
Numeral systems are classified here as to whether they use [[positional notation]] (also known as place-value notation), and further categorized by [[radix]] or base.

===Standard positional numeral systems===
{{unreliable sources|date=April 2023}}
[[File:Binary clock.svg|thumb|A [[binary clock]] might use [[Light-emitting diode|LEDs]] to express binary values. In this clock, each column of LEDs shows a [[binary-coded decimal]] numeral of the traditional [[sexagesimal]] time.]]

The common names are derived [[Hexadecimal#Etymology|somewhat arbitrarily]] from a mix of [[Latin]] and [[Greek language|Greek]], in some cases including roots from both languages within a single name.<ref>For the mixed roots of the word "hexadecimal", see {{citation|title=Discrete Mathematics with Applications|first=Susanna|last=Epp|edition=4th|publisher=Cengage Learning|year=2010|isbn=9781133168669|page=91|url=https://books.google.com/books?id=HUAIAAAAQBAJ&pg=PA91}}.</ref> There have been some proposals for standardisation.<ref>[http://www.dozenal.org/drupal/sites_bck/default/files/MultiplicationSynopsis.pdf Multiplication Tables of Various Bases], p. 45, Michael Thomas de Vlieger, [[Dozenal Society of America]]</ref>
<!--please do not add bases unless you can prove that they have been used FOR MULTIPLE POSITIONS. For example, weeks&days DOES NOT count as base 7, but hours/minutes/seconds DOES count as base 60. Another way to look at this is that there must be a way to count up to at least the square of the base, or else it's not really positional.-->
{| class="wikitable"
|-
! Base !! Name !! Usage
|-

| 2 || [[Binary number|Binary]] || [[Digital signal|Digital]] [[computing]], [[imperial units|imperial]] and [[United States customary units|customary]] [[volume]] ([[bushel]]-[[kenning (unit)|kenning]]-[[peck]]-[[gallon]]-[[English units#Volume|pottle]]-[[quart]]-[[pint]]-[[cup (unit)|cup]]-[[gill (volume)|gill]]-[[English units#Volume|jack]]-[[fluid ounce]]-[[tablespoon]])
|-
| 3 || [[Ternary numeral system|Ternary]], trinary<ref name="Kindra2022">{{Cite journal |last1=Kindra |first1=Vladimir |last2=Rogalev |first2=Nikolay |last3=Osipov |first3=Sergey |last4=Zlyvko |first4=Olga |last5=Naumov |first5=Vladimir |date=2022 |title=Research and Development of Trinary Power Cycles |journal=Inventions |language=en |volume=7 |issue=3 |pages=56 |doi=10.3390/inventions7030056 |doi-access=free |issn=2411-5134}}</ref>|| [[Cantor set]] (all points in [0,1] that ''can'' be represented in ternary with no 1s); [[counting]] [[Tasbih]] in [[Islam]]; [[Hand (unit)|hand]]-[[Foot (unit)|foot]]-[[yard]] and teaspoon-tablespoon-shot [[measurement system]]s; most ''[[radix economy|economical]]'' integer base
|-
| 4 || [[Quaternary numeral system|Quaternary]] || [[Chumashan languages]] and [[Kharosthi#Numerals|Kharosthi numerals]]
|-
| 5 || [[Quinary]] || [[Gumatj language|Gumatj]], [[Ateso language|Ateso]], [[Nunggubuyu language|Nunggubuyu]], [[Kuurn Kopan Noot language|Kuurn Kopan Noot]], and [[Saraveca language|Saraveca]] languages; common count grouping e.g. [[tally marks]]
|-
| 6 || [[Senary]], [[seximal]] || [[Diceware]], [[Ndom language|Ndom]], [[Kanum language|Kanum]], and [[Proto-Uralic language]] (suspected)
|-
| 7 || Septimal, [[Septenary]]<ref>{{Cite web |title=Definition of SEPTENARY |url=https://www.merriam-webster.com/dictionary/septenary |access-date=2023-11-21 |website=www.merriam-webster.com |language=en}}</ref>|| [[Week]]s timekeeping, Western music [[letter notation]]
|-
| 8 || [[Octal]] || [[Charles XII of Sweden#Scientific contributions|Charles XII of Sweden]], [[Filesystem permissions#Octal notation|Unix-like permissions]], [[Transponder (aeronautics)|Squawk codes]], [[PDP-11 architecture#Memory management|DEC PDP-11]], [[Yuki language|Yuki]], [[Pame languages|Pame]], compact notation for binary numbers, Xiantian ([[I Ching]], China)
|-
| 9 || [[Nonary]], [[nonal]] || Compact notation for [[ternary numeral system|ternary]]
|-
| 10 || [[Decimal]], denary || Most widely used by contemporary societies<ref>''The History of Arithmetic'', [[Louis Charles Karpinski]], 200pp, Rand McNally & Company, 1925.</ref><ref>''Histoire universelle des chiffres'', [[Georges Ifrah]], Robert Laffont, 1994.</ref><ref>''The Universal History of Numbers: From prehistory to the invention of the computer'', [[Georges Ifrah]], {{isbn|0-471-39340-1}}, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk</ref>
|-
| 11 || [[Undecimal]], unodecimal, undenary || A base-11 number system was attributed to the [[Māori people|Māori]] ([[New Zealand]]) in the 19th century<ref>{{cite journal |last=Overmann |first=Karenleigh A |date=2020 |title=The curious idea that Māori once counted by elevens, and the insights it still holds for cross-cultural numerical research |url=http://www.thepolynesiansociety.org/jps/index.php/JPS/article/view/458 |journal=Journal of the Polynesian Society |volume=129 |issue=1 |pages=59–84 |doi=10.15286/jps.129.1.59-84 |access-date=24 July 2020|doi-access=free }}</ref> and the [[Pangwa]] ([[Tanzania]]) in the 20th century.<ref name="THOMAS">{{cite journal |last=Thomas |first=N.W |date=1920 |title=Duodecimal base of numeration |journal=Man |volume=20 |issue=1 |pages=56–60 |doi=10.2307/2840036 |jstor=2840036 |url=http://www.jstor.com/stable/2840036 |access-date=25 July 2020}}</ref> Briefly proposed during the [[French Revolution]] to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal. Used as a [[International Standard Book Number#Check digits|check digit in ISBN]] for 10-digit ISBNs. Applications in computer science and technology.<ref>{{cite journal |first=Werner |last= Ulrich |date=November 1957 |title=Non-binary error correction codes |journal=Bell System Technical Journal |volume=36 |issue=6 |pages=1364–1365 |doi= 10.1002/j.1538-7305.1957.tb01514.x |url=https://archive.org/details/bstj36-6-1341/page/n23/mode/2up?q=unodecimal}}</ref><ref>{{cite journal |first1=Debasis |last1=Das |first2=U.A. |last2=Lanjewar |date=January 2012 |title=Realistic Approach of Strange Number System from Unodecimal to Vigesimal |journal=International Journal of Computer Science and Telecommunications |publisher=Sysbase Solution Ltd. |location=London |volume=3 |issue=1 |url=https://www.ijcst.org/Volume3/Issue1/p2_3_1.pdf |page=13}}</ref><ref>{{cite journal|first1=Saurabh |last1=Rawat |first2=Anushree |last2=Sah |date=May 2013 |title=Subtraction in Traditional and Strange Number System by r's and r-1's Compliments |journal=International Journal of Computer Applications |volume=70 |issue=23 |pages=13–17 |doi=10.5120/12206-7640 |bibcode=2013IJCA...70w..13R |quote=... unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed...|doi-access=free }}</ref> Featured in popular fiction.
|-
| 12 || [[Duodecimal]], dozenal || Languages in the [[Nigeria]]n Middle Belt [[Janji language|Janji]], [[Gbiri-Niragu language|Gbiri-Niragu]], [[Piti language|Piti]], and the Nimbia dialect of [[Gwandara language|Gwandara]]; [[Chepang language]] of [[Nepal]], and the Mahl dialect of [[Maldivian language|Maldivian]]; [[dozen]]-[[gross (unit)|gross]]-great gross counting; [[12-hour clock]] and [[month]]s timekeeping; years of [[Chinese zodiac]]; [[Foot (unit)|foot]] and [[inch]]; [[Roman numerals#Fractions|Roman fractions]]; [[penny]] and [[shilling]]
|-
| 13 || Tredecimal, tridecimal{{sfn|Das|Lanjewar|2012|p=13}}{{sfn|Rawat|Sah|2013}} || [[Conway base 13 function]].
|-
| 14 || Quattuordecimal, quadrodecimal{{sfn|Das|Lanjewar|2012|p=13}}{{sfn|Rawat|Sah|2013}} <!--Tetradecimal{{cn|date=April 2023}}--> || Programming for the [[Hewlett-Packard|HP]] 9100A/B calculator<ref>[http://www.hpmuseum.org/prog/hp9100pr.htm HP 9100A/B programming, HP Museum]<!-- https://web.archive.org/web/20180709150317/http://www.hpmuseum.org/prog/hp9100pr.htm --></ref> and image processing applications;<ref>{{Cite web|url=https://www.freepatentsonline.com/6690378.html|title=Image processor and image processing method}}</ref> [[pound (mass)|pound]] and [[stone (mass)|stone]].
|-
| 15 || Quindecimal, pentadecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || Telephony routing over IP, and the [[Huli language]].
|-
| 16 || [[Hexadecimal]], sexadecimal, sedecimal
| Compact notation for [[binary data]]; [[tonal system]]; [[ounce]] and [[pound (mass)|pound]].
|-
| 17 || Septendecimal, heptadecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || 
|-
| 18 || Octodecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}}<!-- duodevicesimal{{cn|date=April 2023}} --> || A base in which 7<sup>n</sup> is [[palindromic number|palindromic]] for n = 3, 4, 6, 9.
|-
| 19 || Undevicesimal, nonadecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}}<!-- enneadecimal{{cn|date=April 2023}} --> || 
|-
| 20 || [[Vigesimal]] || [[Basque language|Basque]], [[Yan Tan Tethera|Celtic]], [[Muisca numerals|Muisca]], [[Inuit numerals|Inuit]], [[Yoruba numerals|Yoruba]], [[Tlingit people|Tlingit]], and [[Dzongkha numerals|Dzongkha]] numerals; [[Santali language|Santali]], and [[Ainu language|Ainu]] languages; [[shilling]] and [[pound (currency)|pound]]
|-
| 5&20 || [[Quinary]]-[[vigesimal]]<ref name="Nykl">{{cite journal |first=Alois Richard |last=Nykl |date=September 1926 |title=The Quinary-Vigesimal System of Counting in Europe, Asia, and America |pages=165–173 |journal=[[Language (journal)|Language]] |volume=2 |issue=3 |url=https://books.google.com/books?id=1GwUAAAAIAAJ&q=Nykl&pg=RA1-PA165 |quote-page=165|quote=A student of the American Indian languages is naturally led to investigate the wide-spread use of the quinary-vigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon.|doi=10.2307/408742 <!--|issn=0097-8507 |eissn=1535-0665 |didn't work for .Raven--> |oclc=50709582 |jstor=408742 |via=[[Google Books]]}}</ref><ref>{{cite book |first=Walter Crosby |last=Eells |chapter=Number Systems of the North American Indians |editor-first1=Marlow |editor-last1=Anderson |editor-first2=Victor |editor-last2=Katz |editor-first3=Robin |editor-last3=Wilson |date=October 14, 2004 |title=Sherlock Holmes in Babylon: And Other Tales of Mathematical History |page=89 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-546-1 |quote='''Quinary-vigesimal'''. This is most frequent. The Greenland Eskimo says 'other hand two' for 7, 'first foot two' for 12, 'other foot two' for 17, and similar combinations to 20, 'man ended.' The Unalit is also quinary to twenty, which is 'man completed.' ... |chapter-url=https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA89 |via=[[Google Books]]}}</ref>{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200 p. 200:] "The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development."}}<!-- pentavigesimal{{cn|date=May 2023}}--> || [[Greenlandic language|Greenlandic]], [[Iñupiaq language#Numerals|Iñupiaq]], [[Kaktovik numerals|Kaktovik]], [[Maya numerals|Maya]], [[Nunivak Cupʼig language|Nunivak Cupʼig]], and [[Central Alaskan Yupʼik|Yupʼik]] numerals – "wide-spread... in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon"<ref name="Nykl"/>
|-
| 21 || <!-- Unvigesimal{{cn|date=April 2023}} --> || The smallest base in which all fractions {{sfrac|1|2}} to {{sfrac|1|18}} have [[repeating decimal|period]]s of 4 or shorter.
|-
| 23 || <!-- trevigesimal, trivigesimal{{cn|date=April 2023}} --> || [[Kalam language]],<ref name="Laycock 1975 219–233">{{cite book |last=Laycock |first=Donald |author-link1=Donald Laycock |date=1975 |editor-last=Wurm |editor-first=Stephen |editor-link1=Stephen Wurm |series=Pacific Linguistics C-38 |title=New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene |publisher=Canberra: Research School of Pacific Studies, Australian National University |pages=219–233 |chapter=Observations on Number Systems and Semantics}}</ref> [[Kobon language]]{{citation needed|date=September 2019|reason=No mention in linked articles, and I only found a few loose mentions; please check if base 23 was actually used in these languages.}}
|-
| 24 || [[:wikt:quadravigesimal|Quadravigesimal]]<ref name="Dibbell">{{cite book |last=Dibbell |first=Julian |chapter=Introduction |title=The Best Technology Writing 2010 |publisher=[[Yale University Press]] |year=2010 |page=9 |isbn=978-0-300-16565-4 |chapter-url=https://books.google.com/books?id=DKPovyrXRwkC&pg=PT9 |quote=There's even a hexavigesimal digital code—our own twenty-six symbol variant of the ancient Latin alphabet, which the Romans derived in turn from the quadravigesimal version used by the ancient Greeks.}}</ref> <!-- tetravigesimal{{cn|date=April 2023}} --> || [[24-hour clock]] timekeeping; [[Greek alphabet]]; [[Kaugel language]].
|-
| 25 || <!-- quinvigesimal, quinquavigesimal, pentavigesimal{{cn|date=April 2023}} --> || Sometimes used as compact notation for quinary.
|-
| 26 || [[:wikt:hexavigesimal|Hexavigesimal]]<ref name="Dibbell"/><ref>{{cite journal|language=en|year=2019 |first1=Brian |last1=Young |first2=Tom |last2=Faris |first3=Luigi |last3=Armogida |title=A nomenclature for sequence-based forensic DNA analysis |publisher=Forensic Science International |journal=Genetics |volume=42 |pages=14–20 |doi=10.1016/j.fsigen.2019.06.001 |pmid=31207427 |url=https://www.sciencedirect.com/science/article/pii/S1872497319300997 |quote=[…] 2) the hexadecimal output of the hash function is converted to hexavigesimal (base-26); 3) letters in the hexavigesimal number are capitalized, while all numerals are left unchanged; 4) the order of the characters is reversed so that the hexavigesimal digits appear […]}}</ref> <!--sexavigesimal{{cn|date=May 2023}}--> || Sometimes used for encryption or ciphering,<ref>{{Cite web|url=http://www.dcode.fr/base-26-cipher|title = Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder}}</ref> using all letters in the [[English alphabet]]
|-
| 27 || <!-- Heptavigesimal,{{cn|date=April 2023}} --> Septemvigesimal ||[[Telefol language|Telefol]],<ref name="Laycock 1975 219–233"/> [[Oksapmin language|Oksapmin]],<ref>{{Cite journal |last1=Saxe |first1=Geoffrey B. |last2=Moylan |first2=Thomas |year=1982 |title=The development of measurement operations among the Oksapmin of Papua New Guinea |journal=Child Development |volume=53 |issue=5 |pages=1242–1248 |doi=10.1111/j.1467-8624.1982.tb04161.x |jstor=1129012}}.</ref> [[Wambon language|Wambon]],<ref>{{cite web | url=https://elementy.ru/problems/264/Bezymyannyy_palets | title=Безымянный палец • Задачи }}</ref> and [[Hewa language|Hewa]]<ref>''[[Nauka i Zhizn]]'', 1992, issue 3, p. 48.</ref> languages. Mapping the nonzero digits to the alphabet and zero to the space is occasionally used to provide [[checksum]]s for alphabetic data such as personal names,<ref>{{citation| last1 = Grannis | first1 = Shaun J.| last2 = Overhage | first2 = J. Marc| last3 = McDonald | first3 = Clement J.| title = Analysis of identifier performance using a deterministic linkage algorithm| pages = 305–309| pmc = 2244404| journal = Proceedings. AMIA Symposium| year = 2002 | pmid=12463836}}.</ref> to provide a concise encoding of alphabetic strings,<ref>{{citation|title=Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code|first=Kenneth Rod|last=Stephens|publisher=Wiley|year=1996|isbn=9780471134183|page=[https://archive.org/details/visualbasicalgor00step/page/215 215]|url=https://archive.org/details/visualbasicalgor00step/page/215}}.</ref> or as the basis for a form of [[gematria]].<ref>{{citation| last = Sallows | first = Lee| issue = 2| journal = [[Word Ways]]| pages = 67–77| title = Base 27: the key to a new gematria| url = http://digitalcommons.butler.edu/wordways/vol26/iss2/2/| volume = 26| year = 1993}}.</ref> Compact notation for [[ternary numeral system|ternary]].
|-
| 28 || <!-- Octovigesimal{{cn|date=April 2023}} --> || Months timekeeping.
|-
| 30 || Trigesimal || The [[Natural Area Code]], this is the smallest base such that all of {{sfrac|1|2}} to {{sfrac|1|6}} terminate, a number n is a [[regular number]] if and only if {{sfrac|1|n}} terminates in base 30.
|-
| [[Base 32|32]] || [[Base32|Duotrigesimal]] || Found in the [[Ngiti language]].
|-
| 33 || <!--Tretrigesimal, tritrigesimal{{cn|date=April 2023}}--> || Use of letters (except I, O, Q) with digits in [[vehicle registration plates of Hong Kong]].
|-
| 34 || <!--Quadritrigesimal, tetratrigesimal{{cn|date=April 2023}}--> || Using all numbers and all letters except I and O; the smallest base where {{sfrac|1|2}} terminates and all of {{sfrac|1|2}} to {{sfrac|1|18}} have periods of 4 or shorter.
|-
| 35 || <!--Pentatrigesimal{{cn|date=April 2023}}--> || Covers the ten decimal digits and all letters of the English alphabet, apart from not distinguishing 0 from O.
|-
| [[Base 36|36]] || [[:wikt:hexatrigesimal|Hexatrigesimal]]<ref>{{cite book |language=en |year=2006 |first=Balázs |last=Gódor |chapter=World-wide user identification in seven characters with unique number mapping |title=Networks 2006: 12th International Telecommunications Network Strategy and Planning Symposium |pages=1–5 |publisher=IEEE |doi=10.1109/NETWKS.2006.300409 |isbn=1-4244-0952-7 |s2cid=46702639 |url=https://ieeexplore.ieee.org/document/4082444 |url-access=subscription |quote=This article proposes the Unique Number Mapping as an identification scheme, that could replace the E.164 numbers, could be used both with PSTN and VoIP terminals and makes use of the elements of the ENUM technology and the hexatrigesimal number system. […] To have the shortest IDs, we should use the greatest possible number system, which is the hexatrigesimal. Here the place values correspond to powers of 36...}}</ref><ref>{{cite journal |language=en |year=2016 |first1=Robert Ssali |last1=Balagadde |first2=Parvataneni |last2=Premchand |title=The Structured Compact Tag-Set for Luganda |journal=International Journal on Natural Language Computing (IJNLC) |volume=5 |issue=4 |issn= |url=https://www.academia.edu/28219615 |quote=Concord Numbers used in the categorisation of Luganda words encoded using either Hexatrigesimal or Duotrigesimal, standard positional numbering systems. […] We propose Hexatrigesimal system to capture numeric information exceeding 10 for adaptation purposes for other Bantu languages or other agglutinative languages.}}</ref> <!--sexatrigesimal{{cn|date=May 2023}}--> || Covers the ten decimal digits and all letters of the [[English alphabet]].
|-
| 37 || <!--septentrigesimal, heptatrigesimal{{cn|date=April 2023}}--> || Covers the ten decimal digits and all letters of the [[Spanish alphabet]].
|-
| 38 || <!--Octotrigesimal{{cn|date=April 2023}}--> || Covers the [[duodecimal]] digits and all letters of the English alphabet.
|-
| 40 || Quadragesimal || [[DEC RADIX 50]]/[[DEC MOD40|MOD40]] encoding used to compactly represent file names and other symbols on [[Digital Equipment Corporation]] computers. The character set is a subset of ASCII consisting of space, upper case letters, the punctuation marks "$", ".", and "%", and the numerals.
|-
| 42 || <!-- Duoquadragesimal{{cn|date=April 2023}}--> || Largest base for which all [[Minimal prime (recreational mathematics)|minimal prime]]s are known.
|-
| 47 || <!-- Septenquadragesimal, septaquadragesimal{{cn|date=April 2023}}--> || Smallest base for which no [[generalized Wieferich prime]]s are known.
|-
| 49 || <!-- novemquadragesimal, enneaquadragesimal{{cn|date=April 2023}}--> || Compact notation for septenary.
|-
| 50 || Quinquagesimal || [[SQUOZE]] encoding used to compactly represent file names and other symbols on some [[IBM]] computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits.
|-
| 52 || <!-- Duoquinquagesimal{{cn|date=April 2023}}--> || Covers the digits and letters assigned to base 62 apart from the basic vowel letters;<ref>{{cite web|url=https://github.com/atomaka/base52|title=Base52|website=[[GitHub]] |access-date=2016-01-03}}</ref> similar  to base 26 but distinguishing upper- and lower-case letters.
|-
| 56 || <!-- Hexaquinquagesimal{{cn|date=April 2023}}--> || A variant of base 58.{{clarify|date=August 2023}}<ref>{{cite web|url=https://gist.github.com/csnover/1110664|title=Base56|access-date=2016-01-03}}</ref>
|-
| 57 || <!-- Heptaquinquagesimal{{cn|date=April 2023}}--> || Covers base 62 apart from I, O, l, U, and u,<ref>{{cite web|url=https://github.com/wyattisimo/base57-ruby|title=Base57|website=[[GitHub]] |access-date=2016-01-03}}</ref> or I, 1, l, 0, and O.<ref>{{cite web|url=https://github.com/skorokithakis/shortuuid|title=Base57|website=[[GitHub]] |access-date=2019-01-22}}</ref>
|-
| 58 || <!-- Octoquinquagesimal{{cn|date=April 2023}}--> || Covers base  62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).<ref name="base58">{{cite web |title=The Base58 Encoding Scheme |url=https://tools.ietf.org/id/draft-msporny-base58 |archive-url=https://web.archive.org/web/20200812103015/https://tools.ietf.org/id/draft-msporny-base58-01.txt |website=[[Internet Engineering Task Force]] |archive-date=August 12, 2020 |date=November 27, 2019 |access-date=August 12, 2020 |quote="Thanks to Satoshi Nakamoto for inventing the Base58 encoding format"}}</ref>
|-
| 60 || [[Sexagesimal]] || [[Babylonian numerals]] and [[Sumerian language|Sumerian]]; [[Degree (angle)|degrees]]-[[Minute and second of arc|minutes-seconds]] and [[hour]]s-[[minute]]s-[[second]]s measurement systems; [[Ekari language|Ekari]]; covers base 62 apart from I, O, and l, but including _(underscore).<ref>{{cite web|url=http://tantek.pbworks.com/w/page/19402946/NewBase60|title=NewBase60|access-date=2016-01-03}}</ref> 
|-
| [[Base62|62]] || <!-- Duosexagesimal{{cn|date=April 2023}}--> || Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet.
|-
| [[Base64|64]] || Tetrasexagesimal || [[I Ching]] in China.<br />This system is conveniently coded into [[ASCII]] by using the 26 letters of the Latin alphabet in both upper and lower case (52 total) plus 10 numerals (62 total) and then adding two special characters (+ and /).
|-
| 72 || <!-- Duoseptuagesimal{{cn|date=April 2023}}--> || The smallest base greater than binary such that no three-digit [[narcissistic number]] exists.
|-
| 80 || Octogesimal || Used as a sub-base in [[Supyire language|Supyire]].
|-
| 85 || <!-- Pentoctogesimal{{cn|date=April 2023}}--> || [[Ascii85]] encoding. This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 85<sup>5</sup> is only slightly bigger than 2<sup>32</sup>. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
|-
| 89 || <!-- Enneaoctogesimal{{cn|date=April 2023}}--> || Largest base for which all [[left-truncatable prime]]s are known.
|-
| 90 || Nonagesimal || Related to [[Goormaghtigh conjecture]] for the generalized [[repunit]] numbers (111 in base 90 = 1111111111111 in base 2).
|-
| 95 || <!-- Pentanonagesimal{{cn|date=April 2023}}--> || Number of <abbr title="incl. the dubiously printable 0x20 space character, which is technically considered part of the set of printable ASCII characters">printable</abbr> ASCII characters.<ref>{{cite web|title=base95 Numeric System|url=http://www.icerealm.org/FTR/?s=docs&p=base95|access-date=2016-01-03|archive-date=2016-02-07|archive-url=https://web.archive.org/web/20160207170735/http://www.icerealm.org/FTR/?s=docs&p=base95|url-status=dead}}</ref>
|-
| 96 || <!-- Hexanonagesimal{{cn|date=April 2023}}--> || Total number of character codes in the (six) [[ASCII stick]]s containing printable characters.
|-
| 97 || <!-- Septanonagesimal{{cn|date=April 2023}}--> || Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized [[Wagstaff prime]]s are known.
|-
| 100 || Centesimal || As 100=10<sup>2</sup>, these are two decimal digits.
|-
| 121 || <!-- Centeunvigesimal{{cn|date=April 2023}}--> || Number expressible with two undecimal digits.
|-
| 125 || <!-- Centepentavigesimal{{cn|date=April 2023}}--> || Number expressible with three quinary digits.
|-
| 128 || <!-- Centeoctovigesimal{{cn|date=April 2023}}--> || Using as 128=2<sup>7</sup>.{{clarify|date=August 2023}}
|-
| 144 || <!-- Centetetraquadragesimal{{cn|date=April 2023}}--> || Number expressible with two duodecimal digits.
|-
| 169 || <!-- Centenovemsexagesimal{{cn|date=April 2023}}--> || Number expressible with two tridecimal digits.
|-
| 185 || <!-- Centepentoctogesimal{{cn|date=April 2023}}--> || Smallest base which is not a [[perfect power]] (where generalized repunits can be factored algebraically) for which no generalized [[repunit prime]]s are known.
|-
| 196 || <!-- Centehexanonagesimal{{cn|date=April 2023}}--> || Number expressible with two tetradecimal digits.
|-
| 210 || <!-- Duocentedecimal{{cn|date=April 2023}}--> || Smallest base such that all fractions {{sfrac|1|2}} to {{sfrac|1|10}} terminate.
|-
| 225 || <!-- Duocentepentavigesimal{{cn|date=April 2023}}--> || Number expressible with two pentadecimal digits.
|-
| 256 || <!-- Duocentehexaquinquagesimal{{cn|date=April 2023}}--> || Number expressible with eight binary digits.
|-
| 360 || <!-- Trecentosexagesimal{{cn|date=April 2023}} --> || [[Degree (angle)|Degree]]s of [[angle]].
|}

===[[Non-standard positional numeral systems]]===

====[[Bijective numeration]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 1 || [[Unary numeral system|Unary{{Spaces|1}}]](Bijective{{Spaces|1}}base{{Non breaking hyphen}}1) || [[Tally mark]]s, [[Counting]]. Unary numbering is used as part of some data compression algorithms such as [[Golomb coding]]. It also forms the basis for the [[Peano axioms]] for formalizing arithmetic within [[mathematical logic]]. A form of unary notation called [[Church encoding]] is used to represent numbers within [[lambda calculus]].
Some [[email]] [[Spam filter|spam filters]] tag messages with a number of [[Asterisk|asterisks]] in an [[e-mail header]] such as ''X-Spam-Bar'' or ''X-SPAM-LEVEL''. The larger the number, the more likely the email is considered spam. 
|-
| 10 || [[Bijective numeration#The bijective base-10 system|Bijective base-10]] ||To avoid [[zero]]
|-
| 26 || [[Bijective numeration#The bijective base-26 system|Bijective base-26]] || [[Spreadsheet]] column [[numeration]]. Also used by [[John Forbes Nash Jr.|John Nash]] as part of his obsession with [[numerology]] and the uncovering of "hidden" messages.<ref>{{cite book | first=Sylvia | last=Nasar | year=2001 | title=A Beautiful Mind | pages=[https://archive.org/details/beautifulmindli00nasa/page/333 333]–6 | publisher=Simon and Schuster | isbn=0-7432-2457-4 | url=https://archive.org/details/beautifulmindli00nasa | url-access=registration }}</ref>
|}

====[[Signed-digit representation]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2 || Balanced binary ([[Non-adjacent form]]) ||
|-
| 3 || [[Balanced ternary]] || [[Ternary computer]]s
|-
| 4 || Balanced quaternary ||
|-
| 5 || Balanced quinary ||
|-
| 6 || Balanced senary ||
|-
| 7 || Balanced septenary ||
|-
| 8 || Balanced octal ||
|-
| 9 || Balanced nonary ||
|-
| 10 || Balanced decimal || [[John Colson]]<br />[[Augustin Cauchy]]
|-
| 11 || Balanced undecimal ||
|-
| 12 || Balanced duodecimal ||
|}

====[[Complex-base system|Complex bases]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2''i'' || [[Quater-imaginary base]] || related to base −4 and [[Hexadecimal|base 16]]
|-
| <math>i\sqrt{2}</math>|| Base <math>i\sqrt{2}</math>|| related to base −2 and [[Quaternary numeral system|base 4]]
|-
| <math>i \sqrt[4]{2}</math>|| Base <math>i \sqrt[4]{2}</math>|| related to [[Binary number|base 2]]
|-
| <math>2 \omega</math> || Base <math>2 \omega</math> || related to [[Octal|base 8]]
|-
| <math>\omega \sqrt[3]{2}</math>|| Base <math>\omega \sqrt[3]{2}</math>|| related to [[Binary number|base 2]]
|-
| −1 ± ''i'' || [[Twindragon]] base || [[Twindragon]] fractal shape, related to base −4 and [[Hexadecimal|base 16]]
|-
| 1 ± ''i'' || Negatwindragon base || related to base −4 and [[Hexadecimal|base 16]]
|}

====[[Non-integer representation|Non-integer bases]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| <math>\frac{3}{2}</math> || Base <math>\frac{3}{2}</math> || a rational non-integer base
|-
| <math>\frac{4}{3}</math> || Base <math>\frac{4}{3}</math> || related to [[duodecimal]]
|-
| <math>\frac{5}{2}</math> || Base <math>\frac{5}{2}</math> || related to [[decimal]]
|-
| [[square root of 2|<math>\sqrt{2}</math>]] || Base <math>\sqrt{2}</math> || related to [[Binary number|base 2]]
|-
| <math>\sqrt{3}</math> || Base <math>\sqrt{3}</math> || related to [[Ternary numeral system|base 3]]
|-
| <math>\sqrt[3]{2}</math> || Base <math>\sqrt[3]{2}</math> ||
|-
| <math>\sqrt[4]{2}</math> || Base <math>\sqrt[4]{2}</math> ||
|-
| [[twelfth root of 2|<math>\sqrt[12]{2}</math>]] || Base <math>\sqrt[12]{2}</math> || usage in [[12 equal temperament|12-tone equal temperament]] musical system
|-
| <math>2\sqrt{2}</math> || Base <math>2\sqrt{2}</math> ||
|-
| <math>-\frac{3}{2}</math> || Base <math>-\frac{3}{2}</math> || a negative rational non-integer base
|-
| <math>-\sqrt{2}</math> || Base <math>-\sqrt{2}</math> || a negative non-integer base, related to [[Binary number|base 2]]
|-
| <math>\sqrt{10}</math> || Base <math>\sqrt{10}</math> || related to [[decimal]]
|-
| <math>2\sqrt{3}</math> || Base <math>2\sqrt{3}</math> || related to [[duodecimal]]
|-
| [[Golden ratio|''φ'']] || [[Golden ratio base]] || early [[Beta encoder]]<ref>
{{Citation
|last= Ward |first=Rachel
|year=2008
|title= On Robustness Properties of Beta Encoders and Golden Ratio Encoders
|journal=[[IEEE Transactions on Information Theory]]
|volume=54 |issue=9 |pages= 4324–4334
|doi=10.1109/TIT.2008.928235
|arxiv=0806.1083|bibcode=2008arXiv0806.1083W
|s2cid=12926540
}}</ref>
|- 
| [[Plastic ratio|''ρ'']] || Plastic number base ||
|-
| [[Supergolden ratio|''ψ'']] || Supergolden ratio base ||
|-
| [[Silver ratio|<math>1+\sqrt{2}</math>]] || Silver ratio base ||
|-
| [[e (mathematical constant)|''e'']] || Base <math>e</math> || best [[radix economy]] {{Citation needed|date=August 2024|reason=This may be true, but without a citation, a layperson has no way of confirming this}}
|-
| [[pi|''π'']] || Base <math>\pi</math> ||
|-
| {{mvar|[[e (mathematical constant)|e]]}}[[Pi|{{pi}}]] || Base <math>e\pi</math> ||
|-
| <math>e^\pi</math> || Base <math>e^\pi</math> ||
|}

====[[p-adic number|''n''-adic number]]====

{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2 || Dyadic number ||
|-
| 3 || Triadic number ||
|-
| 4 || Tetradic number || the same as dyadic number
|-
| 5 || Pentadic number ||
|-
| 6 || Hexadic number || not a [[field (mathematics)|field]]
|-
| 7 || Heptadic number ||
|-
| 8 || Octadic number || the same as dyadic number
|-
| 9 || Enneadic number || the same as triadic number
|-
| 10 || Decadic number || not a field
|-
| 11 || Hendecadic number ||
|-
| 12 || Dodecadic number || not a field
|}

====[[Mixed radix]]====
* [[Factorial number system]] {1, 2, 3, 4, 5, 6, ...}
* Even double factorial number system {2, 4, 6, 8, 10, 12, ...}
* Odd double factorial number system {1, 3, 5, 7, 9, 11, ...}
* [[Mixed radix#Primorial number system|Primorial number system]] {2, 3, 5, 7, 11, 13, ...}
* [[Fibonorial]] number system {1, 2, 3, 5, 8, 13, ...}
* {60, 60, 24, 7} in timekeeping
* {60, 60, 24, 30 (or 31 or 28 or 29), 12, 10, 10, 10} in timekeeping
* (12, 20) traditional English monetary system (£sd)
* (20, 18, 13) Maya timekeeping

====Other====
* [[Quote notation]]
* [[Redundant binary representation]]
* [[Goodstein's theorem#Hereditary base-n notation|Hereditary base-n notation]]
* [[Asymmetric numeral systems]] optimized for non-uniform probability distribution of symbols
* [[Combinatorial number system]]

===Non-positional notation===
All known numeral systems developed before the [[Babylonian numerals]] are non-positional,{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=kXZhBAAAQBAJ&pg=PA254 p. 254:] Chrisomalis calls the Babylonian system "the first positional system ever"}} as are many developed later, such as the [[Roman numerals]].  The French Cistercian monks created [[Cistercian numerals|their own numeral system.]]

==See also==

{{col div|colwidth=40em}}
* {{annotated link|History of ancient numeral systems}}
* {{annotated link|History of the Hindu–Arabic numeral system}}
* {{annotated link|List of numeral system topics}}
* {{annotated link|Numeral prefix}}
* {{annotated link|Radix}}
* {{annotated link|Radix economy}} 
* {{annotated link|Timeline of numerals and arithmetic}}

{{colend}}

==References==
{{Reflist}}

[[Category:Numeral systems| ]]
[[Category:Number-related lists|Systems]]
[[Category:Lists of writing systems|Numeral]]