List of numeral systems

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

By culture

Name Base Sample Approx. first appearance
Babylonian numerals 60 3100 BC
Egyptian numerals 10
Z1
V20
V1
M12
D50
I8

or
I7
C11
3000 BC
Maya numerals 20
Indian Numerals 10 0 १ २ ३ ४ ५ ६ ७ ८ ९ 750 BC – 690 BC
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) 10 ○/零 一 二 三 四 五 六 七 八 九
Roman numerals 10 Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ 1000 BC
Greek numerals 10 α β γ δ ε ϝ ζ η θ ι After 100 BC
Chinese rod numerals 10 1st century
Hindu-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

By type of notation

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

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
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 Celtic, Maya, 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 Septemvigesimal 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 Duopentagesimal Base52 encoding, a variant of Base62 without vowels[11]
56 Hexapentagesimal Base56 encoding, a variant of Base58[12]
57 Heptapentagesimal Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[13]
58 Octopentagesimal 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
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]

Non-standard positional numeral systems

Bijective numeration

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

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

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

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

Non-integer bases

Base Name Usage
φ Golden ratio base Early Beta encoder[21]
e Base e Lowest radix economy
π Base \pi "Pi-nary"
√2 Base \sqrt{2}
¹²√2 Base \sqrt[12]{2} Scientific pitch notation

Other

Non-positional notation

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

See also

References

  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. ↑ http://www.dcode.fr/base-26-cipher
  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 2244404, 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.
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