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| Numeral systems |
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| List of numeral systems |
There are many different numeral systems, that is, writing systems for expressing numbers.
"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system.": 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. 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).
| Name | Base | Sample | Approx. First Appearance | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Proto-cuneiform numerals | 10&60 | c. 3500–2000 BCE | ||||||||||
| Indus numerals | unknown | c. 3500–1900 BCE | ||||||||||
| Proto-Elamite numerals | 10&60 | 3100 BCE | ||||||||||
| Sumerian numerals | 10&60 | 3100 BCE | ||||||||||
| Egyptian numerals | 10 |
| 3000 BCE | |||||||||
| Babylonian numerals | 10&60 |           | 2000 BCE | |||||||||
| Aegean numerals | 10 | 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏 (          )𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘 (          )𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 (          )𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪 (          )𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 (          ) | 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) |
1300 BCE | |||||||||
| Roman numerals | 5&10 | I V X L C D M | 1000 BCE | |||||||||
| Attic numerals | 5&10 | Ι 𐅃 Δ 𐅄 Η 𐅅 Χ 𐅆 Μ 𐅇 | 7th century 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 | <400 BCE | |||||||||
| Kharosthi numerals | 4&10 | 𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀 | <400–250 BCE | |||||||||
| Maya numerals | 5&20 |                     𝋠 𝋡 𝋢 𝋣 𝋤 𝋥 𝋦 𝋧 𝋨 𝋩 𝋪 𝋫 𝋬 𝋭 𝋮 𝋯 𝋰 𝋱 𝋲 𝋳 | 3rd century BCE | |||||||||
| Phoenician numerals | 10 | 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 | <250 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 ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ |
3rd century BCE (Brahmi) | |||||||||
| Hebrew numerals | 10 | 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 | 1st century BCE | |||||||||
| Chinese rod numerals | 10 | 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 | 1st century | |||||||||
| Coptic numerals | 10 | Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ | 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 | 3rd–4th century 15th century (Modern Style): 135–136 | |||||||||
| Armenian numerals | 10 | Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ | Early 5th century | |||||||||
| Khmer numerals | 10 | ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ | Early 7th century | |||||||||
| Thai numerals | 10 | ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ | 7th century | |||||||||
| 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 | <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) | late 7th/early 8th century | |||||||||
| Eastern Arabic numerals | 10 | ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ ۹ ۸ ۷ ۶ ۵ ۴ ۳ ۲ ۱ ۰ (Persian) | 8th century | |||||||||
| Vietnamese numerals (Chữ Nôm) | 10 | 𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩 | <9th century | |||||||||
| Western Arabic numerals | 10 | 0 1 2 3 4 5 6 7 8 9 | 9th century | |||||||||
| Glagolitic numerals | 10 | Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... | 9th century | |||||||||
| Cyrillic numerals | 10 | а в г д е ѕ з и ѳ і ... | 10th century | |||||||||
| Rumi numerals | 10 |  | 10th century | |||||||||
| Burmese numerals | 10 | ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ | 11th century | |||||||||
| Tangut numerals | 10 | 𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗 | 11th century (1036) | |||||||||
| Cistercian numerals | 10 |  | 13th century | |||||||||
| Muisca numerals | 20 |  | 15th century | |||||||||
| Korean numerals (Hangul) | 10 | 영 일 이 삼 사 오 육 칠 팔 구 | 15th century (1443) | |||||||||
| Aztec numerals | 20 |        (1, 5, 20, 100, 400, 800, 8000) | 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 | 18th century | |||||||||
| Pentadic runes | 10 |  | 19th century | |||||||||
| Cherokee numerals | 10 | .png){kind=small} | 19th century (1820s) | |||||||||
| Vai numerals | 10 | ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ | 19th century (1832) | |||||||||
| Bamum numerals | 10 | ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ | 19th century (1896) | |||||||||
| Mende Kikakui numerals | 10 | 𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 | 20th century (1917) | |||||||||
| Osmanya numerals | 10 | 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 | 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 | 20th century (1930s) | |||||||||
| N'Ko numerals | 10 | ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ | 20th century (1949) | |||||||||
| Hmong numerals | 10 | 𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭙 𖭑𖭐 | 20th century (1959) | |||||||||
| Garay numerals | 10 |  | 20th century (1961) | |||||||||
| Adlam numerals | 10 | 𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 | 20th century (1989) | |||||||||
| Kaktovik numerals | 5&20 |                     𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 | 20th century (1994) | |||||||||
| Sundanese numerals | 10 | ᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹ | 20th century (1996) |
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.
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. There have been some proposals for standardisation.
| 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 | 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), 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). |
| 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. |
| 11 | Undecimal, unodecimal, undenary | A base-11 number system was mistakenly attributed to the Māori (New Zealand) in the 19th century and one was reported to be used by the Pangwa (Tanzania) in the 20th century, but was not confirmed by later research and is believed to also be an error. 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. |
| 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 | Conway's base 13 function. |
| 14 | Quattuordecimal, quadrodecimal | Programming for the HP 9100A/B calculator and image processing applications. |
| 15 | Quindecimal, pentadecimal | Telephony routing over IP, and the Huli language. |
| 16 | Hexadecimal, sexadecimal, sedecimal | Compact notation for binary data; tonal system of Nystrom. |
| 17 | Heptadecimal, septendecimal | |
| 18 | Octodecimal | |
| 19 | Undevicesimal, nonadecimal | |
| 20 | Vigesimal | Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages. |
| 5&20 | Quinary-vigesimal | 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." |
| 23 | Kalam language, | |
| 24 | Quadravigesimal | 24-hour clock timekeeping; Greek alphabet; Kaugel language. |
| 26 | Hexavigesimal | Sometimes used for encryption or ciphering, 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) and SID (sequence identification, an indexing system of PCR amplicons in forensics). |
| 27 | Telefol, Oksapmin, Wambon, and Hewa 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, to provide a concise encoding of alphabetic strings, or as the basis for a form of gematria. Compact notation for ternary. | |
| 30 | Trigesimal | |
| 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 RFC 4648, with each character standing for 5 bits. |
| 36 | Hexatrigesimal | Used to encode large numbers into an alphanumeric string (26 letters, 10 numbers). |
| 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. | |
| 50 | SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. | |
| 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). |
| 64 | Used to encode computer (binary) data into a relatively compact string, with each character standing for 6 bits (RFC 4648). | |
| 80 | Used as a sub-base in Supyire. | |
| 90 | Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2). |
Unary, or bijective base‑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. [citation needed]
| Base | Name | Usage |
|---|---|---|
| 2 | Balanced binary (Non-adjacent form) | |
| 3 | Balanced ternary | Ternary computers |
| 10 | Balanced decimal | John Colson Augustin Cauchy |
| Base | Name | Usage |
|---|---|---|
| 2i | Quater-imaginary base | related to base −4 and base 16 |
| −1 ± i | Twindragon base | Twindragon fractal shape, related to base −4 and base 16 |
| Base | Name | Usage |
|---|---|---|
| φ | Golden ratio base | early Beta encoder |
| e | Base | best radix economy [citation needed] |
![[icon]](https://en.wikipedia.org/wiki/File:Wiki_letter_w_cropped.svg){kind=small} | This section needs expansion. You can help by adding missing information. (March 2026) |
All known numeral systems developed before the Babylonian numerals are non-positional, as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.
The Hebrew as well as the Greek date 25, which is the 25th year of reign of Alexander Yannai (78 B.C.E.), were recently discerned.
{{citation}}: CS1 maint: work parameter with ISBN (link)... unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed...
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.
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.' ...
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.
[…] 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 […]
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...
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.