List of numeral systems

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This is a list of numeral systems.

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
Z1
V20
V1
M12
D50
I8

or
I7
C11
3000 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
Oracle bone script 0 0 14th century 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 Counting rod v1.png Counting rod v2.png Counting rod v3.png Counting rod v4.png Counting rod v5.png Counting rod v6.png Counting rod v7.png Counting rod v8.png Counting rod v9.png Counting rod h1.png 1st century
Hindu-Arabic Numerals 10 0 1 2 3 4 5 6 7 8 9 9th century
John Napier's Location arithmetic 2 a b ab c ac bc abc d ad bd 1617 in Rabdology, a non-positional binary system

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 language, and Proto-Uralic language (suspected)
7 Septenary Week cycle
8 Octal Charles XII of Sweden, Unix-like permissions, DEC PDP-11, compact notation for binary numbers
9 Nonary compact notation of ternary 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 A cycle of the Maya calendar
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 or quaternary numbers; tonal system
18 Octodecimal A cycle of the Mesoamerican Long Count calendar
20 Vigesimal Celtic, Maya, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
24 Tetravigesimal Kaugel language; hours timekeeping
25 Pentavigesimal Compact notation of quinary numbers
26 Hexavigesimal Uses of letters without digits, e.g. spreadsheet column numeration
27 Septemvigesimal Telefol and Oksapmin languages; compact notation of ternary numbers
28 Octovigesimal Four week month of thirteen month calendar
30 Trigesimal Month cycle for various calendars; The Natural Area Code
32 Duotrigesimal Base32 encoding; compact notation of binary; and the Ngiti language
36 Hexatrigesimal Base36 encoding; use of letters with digits; compact notation of senary numbers
60 Sexagesimal Babylonian numerals; degrees-minutes-seconds and hours-minutes-seconds measurement systems
62 Duosexagesimal Base62 encoding; using all English letters (capital and lowercase) and digits but no others, e.g. URL shortening.
64 Tetrasexagesimal Base64 encoding; compact notation of binary, quaternary or octal numbers
85 Pentaoctagesimal Ascii85 encoding
120 Centovigesimal Great hundred
240 Duocentoquadragesimal £sd
256 Internally in computers
360 Trecentosexagesimal Degree division of circle

Non-standard positional numeral systems[edit]

Bijective numeration[edit]

Base Name Usage
10 Bijective base-10
26 Bijective base-26 Spreadsheet column numeration

Signed-digit representation[edit]

Base Name Usage
2 Non-adjacent form
3 Balanced ternary Ternary computers
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[7]
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[edit]

Non-positional notation[edit]

All known numeral systems developed before the Babylonian numerals are non-positional.[8]

1-adic bijective numeration[edit]

Base Name Usage
1 Unary Tally marks

See also[edit]

References[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. ^ 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 
  8. ^ 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 .