List of numeral systems

{{short description|none}} {{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}} {{Contains special characters | special = uncommon Unicode characters | fix = Help:Multilingual support | error = question marks, boxes, or other symbols | characters = the intended characters | image = Replacement character.svg | link = Specials (Unicode block)#Replacement character | alt = <?> | compact = }} {{Numeral systems}}

There are many different numeral systems, that is, writing systems 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 skin-invert-image" |- ! 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 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 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 || 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px || {{sort|-2000|2000 BCE}} |- | Aegean numerals || 10 || 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  ( 10px|frameless|1 10px|frameless|2 10px|frameless|3 10px|frameless|4 10px|frameless|5 10px|frameless|6 10px|frameless|7 10px|frameless|8 10px|frameless|9 ) 𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘  ( 10px|frameless|10 10px|frameless|20 10px|frameless|30 10px|frameless|40 10px|frameless|50 10px|frameless|60 10px|frameless|70 10px|frameless|80 10px|frameless|90 ) 𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡  ( 10px|frameless|100 10px|frameless|200 10px|frameless|300 10px|frameless|400 10px|frameless|500 10px|frameless|600 10px|frameless|700 10px|frameless|800 10px|frameless|900 ) 𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪  ( 10px|frameless|1000 10px|frameless|2000 10px|frameless|3000 10px|frameless|4000 10px|frameless|5000 10px|frameless|6000 10px|frameless|7000 10px|frameless|8000 10px|frameless|9000 ) 𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳  ( 10px|frameless|10000 10px|frameless|20000 10px|frameless|30000 10px|frameless|40000 10px|frameless|50000 10px|frameless|60000 10px|frameless|70000 10px|frameless|80000 10px|frameless|90000 )|| {{sort|-1500|1500 BCE}} |- |Chinese numerals Japanese numerals Korean numerals (Sino-Korean) Vietnamese numerals (Sino-Vietnamese)|| 10 || 零一二三四五六七八九十百千萬億 (Default, Traditional Chinese) 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 י כ ל מ נ ס ע פ צ ק ר ש ת ך ם ן ף ץ}} || {{sort|-800|800 BCE}} |- | Indian numerals || 10 ||

Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯

Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९

Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯

Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯

Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯

Odia ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯

Punjabi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯

Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯

Tibetan <big>༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩</big>

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 ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ || {{sort|-400|<400 BCE}} |- |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.org}}</ref> |- | Phoenician numerals || 10 || 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 <ref>{{cite web |last1=Everson|first1=Michael |author-link=Michael Everson |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 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 ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ 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}} 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}} |- | Maya numerals || 5{{resize|88%|&}}20 || 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 𝋠 𝋡 𝋢 𝋣 𝋤 𝋥 𝋦 𝋧 𝋨 𝋩 𝋪 𝋫 𝋬 𝋭 𝋮 𝋯 𝋰 𝋱 𝋲 𝋳 || {{sort|320|4th century}} |- | 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) 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 numerals || 10 || 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 || frameless|upright || {{sort|1200|13th century}} |- | Muisca numerals || 20 || 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 || x25px x25px x25px x25px x25px x25px x30px (1, 5, 20, 100, 400, 800, 8000)|| {{sort|1500|16th century}} |- | Sinhala numerals || 10 || |෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣 𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴 || {{sort|1699|18th century}} |- | Pentadic runes || 10 || frameless|upright || {{sort|1800|19th century}} |- | Cherokee numerals || 10 || frameless|upright=2 || {{sort|1820|19th century (1820s)}} |- | 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 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 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 numerals || 10 || 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 || {{sort|1921|20th century (1920s)}} |- | 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 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>}} |- | 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 numerals || 10 || 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 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}} 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> |}

==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 LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.|class=skin-invert-image]]

The common names are derived somewhat arbitrarily from a mix of Latin and 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>  {| class="wikitable" |- ! Base !! Name !! Usage |-

| 2 || Binary || Digital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon). |- | 3 || 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-foot-yard and teaspoon-tablespoon-shot measurement systems; most ''economical'' integer base. |- | 4 || Quaternary || Chumashan languages and Kharosthi numerals. |- | 5 || Quinary || Aneityum (traditional),<ref> The Aneityum language of Aneityum, Vanuatu traditionally had a quinary system, although this had largely been replaced by a decimal system, based on English numbers, by the early 20th century.</ref> Ateso, Gumatj, Kuurn Kopan Noot, and Nunggubuyu, Saraveca languages; common count grouping e.g. tally marks. |- | 6 || Senary, seximal || Diceware, Ndom, Kanum, and Proto-Uralic language (suspected). |- | 7 || Septimal, septenary || |- | 8 || Octal || Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China). |- | 9 || Nonary, nonal || Compact notation for 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 mistakenly attributed to the Māori (New Zealand) in the 19th century<ref name="Overmann">{{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 one was reported to be used by 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> but was not confirmed by later research and is believed to also be an error.<ref name="Hammarström">{{cite book |last1=Hammarström |first1=Harald |chapter=Rarities in numeral systems |title=Rethinking Universals|isbn=9783110220933 |year=2010 |pages=11–60 |doi=10.1515/9783110220933.11}}</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 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> |- | 12 || Duodecimal, dozenal || Languages in the Nigerian Middle Belt Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozen-gross-great gross counting; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions. |- | 13 || Tredecimal, tridecimal{{sfn|Das|Lanjewar|2012|p=13}}{{sfn|Rawat|Sah|2013}} || Conway's base 13 function. |- | 14 || Quattuordecimal, quadrodecimal{{sfn|Das|Lanjewar|2012|p=13}}{{sfn|Rawat|Sah|2013}}  || Programming for the HP 9100A/B calculator<ref>[http://www.hpmuseum.org/prog/hp9100pr.htm HP 9100A/B programming, HP Museum]</ref> and image processing applications.<ref>{{Cite web|url=https://www.freepatentsonline.com/6690378.html|title=Image processor and image processing method}}</ref> |- | 15 || Quindecimal, pentadecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || Telephony routing over IP, and the Huli language.<ref name="Hammarström"/> |- | 16 || Hexadecimal, sexadecimal, sedecimal | Compact notation for binary data; tonal system of Nystrom. |- | 17 || Heptadecimal, septendecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || |- | 18 || Octodecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || |- | 19 || Undevicesimal, nonadecimal{{sfn|Das|Lanjewar|2012|p=14}}{{sfn|Rawat|Sah|2013}} || |- | 20 || Vigesimal || Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages. |- | 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 |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  |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 |editor2-link=Victor J. Katz |editor-first3=Robin |editor-last3=Wilson |editor3-link=Robin Wilson (mathematician) |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."}} || Greenlandic, Iñupiaq, Kaktovik, Maya, Nunivak Cupʼig, and Yupʼik numerals – "wide-spread... in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon."<ref name="Nykl"/> |- | 21 || || The smallest base in which all fractions {{sfrac|1|2}} to {{sfrac|1|18}} have periods of 4 or shorter. |- | 23 || || 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> |- | 24 || 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>  || 24-hour clock timekeeping; Greek alphabet; Kaugel language. |- | 25 || || Compact notation for quinary. |- | 26 || Hexavigesimal<ref name="Dibbell"/><ref name=byoung19>{{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 […]|url-access=subscription }}</ref>  || 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. Used to encode SHA-256 hashes into uppercase letters in InChIKey (a standard indexing system of chemical structures)<ref>{{cite web |url=http://www.inchi-trust.org/technical-faq/#13.1 |title=Technical FAQ - InChI Trust |website=inchi-trust.org |access-date=2021-01-08}}</ref> and SID (sequence identification, an indexing system of PCR amplicons in forensics).<ref name=byoung19/> |- | 27 || ||Telefol,<ref name="Laycock 1975 219–233"/> 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,<ref>{{cite web | url=https://elementy.ru/problems/264/Bezymyannyy_palets | title=Безымянный палец • Задачи }}</ref> and 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 checksums 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. |- | 30 || Trigesimal<ref>{{cite web |author= |date= December 2024 |title= trigesimal - adjective |url= https://www.oed.com/dictionary/trigesimal_adj |website= Oxford English Dictionary |location= Oxford, England |publisher= Oxford University Press |access-date= 2026-04-26 }}</ref> || 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. |- | 32 || Duotrigesimal || Found in the Ngiti language. Also used to encode computer (binary) data into an alphanumerical string without confusable characters (e.g. zero and "O", eight and "B") in {{IETF RFC|4648}}, with each character standing for 5 bits. |- | 34 || || The smallest base where {{sfrac|1|2}} terminates and all of {{sfrac|1|2}} to {{sfrac|1|18}} have periods of 4 or shorter. |- | 36 || 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 |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 |volume=5 |issue=4 |pages=01–21 |doi=10.5121/ijnlc.2016.5401 |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>  || Used to encode large numbers into an alphanumeric string (26 letters, 10 numbers). Compact notation for senary. |- | 40 || || DEC RADIX 50/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 || || Largest base for which all minimal primes are known. |- | 47 || || Smallest base for which no generalized Wieferich primes are known. |- | 50 || || SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits. |- | 60 || Sexagesimal || Babylonian numerals and Sumerian; degrees-minutes-seconds and hours-minutes-seconds measurement systems; 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> |- | 64 || || Used to encode computer (binary) data into a relatively compact string, with each character standing for 6 bits ({{IETF RFC|4648}}). |- | 72 || || The smallest base greater than binary such that no three-digit narcissistic number exists. |- | 80 || || Used as a sub-base in Supyire. |- | 89 || || Largest base for which all left-truncatable primes are known. |- | 90 || || Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2). |- | 97 || || Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff primes are known. |- | 185 || || Smallest base which is not a perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known. |- | 210 || || Smallest base such that all fractions {{sfrac|1|2}} to {{sfrac|1|10}} terminate. |}

===Non-standard positional numeral systems===

====Bijective numeration==== Unary, or bijective base{{Non breaking hyphen}}1, is used in Tally marks, and 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 filters tag messages with a number of 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. {{cn|date=May 2026}}

====Signed-digit representation==== {| class="wikitable" |- ! Base !! Name !! Usage |- | 2 || Balanced binary (Non-adjacent form) || |- | 3 || Balanced ternary || Ternary computers |- | 10 || Balanced decimal || John Colson Augustin Cauchy |- |}

====Complex bases==== {| class="wikitable" |- ! Base !! Name !! Usage |- | 2''i'' || Quater-imaginary base || related to base −4 and base 16 |- | −1 ± ''i'' || Twindragon base || Twindragon fractal shape, related to base −4 and base 16 |}

====Non-integer bases==== {| class="wikitable" |- ! Base !! Name !! Usage |- | ''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> |- | ''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}} |}

====Mixed radix==== * Factorial number system {1, 2, 3, 4, 5, 6, ...} * Primorial number system {2, 3, 5, 7, 11, 13, ...}

====Other==== * Quote notation * Redundant binary representation * Hereditary base-n notation * Asymmetric numeral systems optimized for non-uniform probability distribution of symbols * Combinatorial number system

===Non-positional notation=== {{expand section|date=March 2026}}

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 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 books on history of number systems}} * {{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}}

==References== {{Reflist}}

Category:Numeral systems Systems Numeral