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	or	3000 BC
Maya numerals	20		<15th century
Muisca numerals	20		<15th century
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	Before 5th century 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, 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	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 85^5 is only slightly bigger than 2^32. 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

• Mixed radix
• Quote notation
• Redundant binary representation

Non-positional notation

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

See also

• Radix
• Radix economy
• Table of bases
• List of numbers in various languages (cardinal number names)
• Numeral prefix

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.