List of numeral systems

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This is a list of numeral systems, that is, writing systems for expressing numbers.

By culture[edit]

Name Base Sample Approx. first appearance
Babylonian numerals 60 Babylonian 1.svgBabylonian 2.svgBabylonian 3.svgBabylonian 4.svgBabylonian 5.svgBabylonian 6.svgBabylonian 7.svgBabylonian 8.svgBabylonian 9.svgBabylonian 10.svg 3100 BC
Egyptian numerals 10

3000 BC
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
c1500 BC
Maya numerals 20 0 maia.svg 1 maia.svg 2 maia.svg 3 maia.svg 4 maia.svg 5 maia.svg 6 maia.svg 7 maia.svg 8 maia.svg 9 maia.svg 10 maia.svg 11 maia.svg 12 maia.svg 13 maia.svg 14 maia.svg 15 maia.svg 16 maia.svg 17 maia.svg 18 maia.svg 19 maia.svg <15th century
Muisca numerals 20 Muisca cyphers acc acosta humboldt zerda.svg <15th century
Indian Numerals 10 Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Devanagari 0 १ २ ३ ४ ५ ६ ७ ८ ९

750 BC – 690 BC
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) 10 〇/零 一 二 三 四 五 六 七 八 九 十
Chinese rod numerals 10 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 1st century
Roman numerals 10 N
1000 BC
Greek numerals 10 ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ
Before 5th century BC
Western Arabic numerals 10 0 1 2 3 4 5 6 7 8 9 9th century
Thai numerals 10 ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙
John Napier's Location arithmetic 2 a b ab c ac bc abc d ad bd abd cd acd bcd abcd 1617 in Rabdology, a non-positional binary system
Hebrew numerals 10 א ב ג ד ה ו ז ח ט
י כ ל מ נ ס ע פ צ
800 BC
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

By type of notation[edit]

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[edit]

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.

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.[1]

Base Name Usage
2 Binary Digital computing
3 Ternary 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 Data transmission and Hilbert curves; Chumashan languages, and Kharosthi numerals
5 Quinary Gumatj, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6 Senary Diceware, Ndom, Kanum, and Proto-Uralic language (suspected)
8 Octal Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian (I Ching, China)
10 Decimal Most widely used by modern civilizations[2][3][4]
11 Undecimal Jokingly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal
12 Duodecimal 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; hours and months timekeeping; years of Chinese zodiac; foot and inch.
13 Tridecimal Conway base 13 function
14 Tetradecimal Programming for the HP 9100A/B calculator[5] and image processing applications[6]
15 Pentadecimal Telephony routing over IP, and the Huli language
16 Hexadecimal Base16 encoding; compact notation for binary data; tonal system
20 Vigesimal Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
24 Tetravigesimal Kaugel language
26 Hexavigesimal Base 26 encoding; sometimes used for encryption or ciphering.[7]
27 Heptavigesimal 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,[8] to provide a concise encoding of alphabetic strings,[9] or as the basis for a form of gematria.[10]
30 Trigesimal The Natural Area Code
32 Duotrigesimal Base32 encoding and the Ngiti language
36 Hexatrigesimal Base36 encoding; use of letters with digits
52 Duoquinquagesimal Base52 encoding, a variant of Base62 without vowels[11]
56 Hexaquinquagesimal Base56 encoding, a variant of Base58[12]
57 Heptaquinquagesimal Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[13]
58 Octoquinquagesimal Base58 encoding
60 Sexagesimal Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[14] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
61 Unsexagesimal NewBase61 encoding, variant of NewBase60 with a space[15]
62 Duosexagesimal Base62 encoding, using 0-9, A-Z, and a-z
64 Tetrasexagesimal Base64 encoding, , I Ching in China
65 Pentasexagesimal Base65 encoding, a variant of Base64[16]
85 Pentoctogesimal 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 855 is only slightly bigger than 232. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
91 Unnonagesimal Base91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92 Duononagesimal Base92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[17]
94 Tetranonagesimal Base94 encoding, using all of ASCII printable characters.[18]
95 Pentanonagesimal Base95 encoding, a variant of Base94 with the addition of the Space character.[19]
256 Ducentahexaquinquagesimal Base256 encoding

Non-standard positional numeral systems[edit]

Bijective numeration[edit]

Base Name Usage
1 Unary (Bijective base-1) Tally marks
10 Bijective base-10
26 Bijective base-26 Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.[20]

Signed-digit representation[edit]

Base Name Usage
2 Balanced binary (Non-adjacent form)
3 Balanced ternary Ternary computers
5 Balanced quinary
9 Balanced nonary
10 Balanced decimal John Colson
Augustin Cauchy

Negative bases[edit]

The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:

Base Name Usage
−2 Negabinary
−3 Negaternary
−10 Negadecimal

Complex bases[edit]

Base Name Usage
2i Quater-imaginary base
−1 ± i Twindragon base Twindragon fractal shape

Non-integer bases[edit]

Base Name Usage
φ Golden ratio base Early Beta encoder[21]
e Base Lowest radix economy


Non-positional notation[edit]

All known numeral systems developed before the Babylonian numerals are non-positional.[22]


  • In this Youtube video, Matt Parker jokingly invented a base-1082 system (From a Fermi estimate of the number of atoms in the universe). He uses this numeral system to describe how many atom carry-overs (i.e. number of parallel universes, or digits in this radix) it takes to have a hypothetical supercomputer generate all possible 256×256 grayscale images, when the root universe lasts 1017 seconds (another Fermi estimate). This turns out to be 1925.

See also[edit]


  1. ^ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669 .
  2. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  3. ^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
  4. ^ 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
  5. ^ HP Museum
  6. ^ Free Patents Online
  7. ^
  8. ^ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proc AMIA Symp., pp. 305–309, PMC 2244404Freely accessible, PMID 12463836 .
  9. ^ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183 .
  10. ^ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77 .
  11. ^ "Base52". Retrieved 2016-01-03. 
  12. ^ "Base56". Retrieved 2016-01-03. 
  13. ^ "Base57". Retrieved 2016-01-03. 
  14. ^ "NewBase60". Retrieved 2016-01-03. 
  15. ^ "NewBase61". Retrieved 2016-01-03. 
  16. ^ "Base65 Encoding". Retrieved 2016-01-03. 
  17. ^ "Base92". Retrieved 2016-01-03. 
  18. ^ "Base94". Retrieved 2016-01-03. 
  19. ^ "base95 Numeric System". Retrieved 2016-01-03. 
  20. ^ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4. 
  21. ^ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, doi:10.1109/TIT.2008.928235 
  22. ^ Chrisomalis calls the Babylonian system "the first positional system ever" in Chrisomalis, Stephen (2010), Numerical Notation: A Comparative History, Cambridge University Press, p. 254, ISBN 9781139485333 .