List of numeral systems

Arabic, Eastern Arabic, Roman, Bengali–Assamese, Malayalam, Thai, and Chinese numerals

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

By culture / time period

NameBaseSampleApprox. first appearance
Prehistoric numerals35,000 BC
Babylonian numerals603100 BC
Egyptian numerals10
Z1
V20
V1
M12
D50
I8

or
I7
C11
3000 BC
Aegean numerals10𐄇 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
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean)10〇/零 一/壹 二/貳 三/叄 四/䦉 五/伍 六/陸 七/柒 八/捌 九/玖 十/拾
Roman numerals10Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ
L C D M		1000 BC
Hebrew numerals10א ב ג ד ה ו ז ח ט
י כ ל מ נ ס ע פ צ		800 BC
Indian Numerals10Tamil 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
Greek numerals10ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ		Before 5th century BC
Cyrillic numerals10А҃ В҃ Г҃ Д҃ Е҃ Ѕ҃ З҃ И҃ Ѳ҃ І҃ ...10th century
Ge'ez numerals-፩, ፪, ፫, ፬, ፭, ፮, ፯, ፰, ፱
፲, ፳, ፴, ፵, ፶, ፷, ፸, ፹, ፺, ፻		3rd-4th century CE, modern style from 15th century CE[1]
Chinese rod numerals10𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩1st century
Phoenician numerals10𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [2]Before 250 AD[3]
Thai numerals10๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙7th century[4]
Abjad numerals10غ 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 اbefore 8th century
Eastern Arabic numerals 10 ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ 8th century
Western Arabic numerals100 1 2 3 4 5 6 7 8 99th century
Burmese numerals 10 ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ 11th century[5]
Maya numerals20 <15th century
Muisca numerals20<15th century
Aztec numerals2016th century
John Napier's Location arithmetic2a b ab c ac bc abc d ad bd abd cd acd bcd abcd1617 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.[6] There have been some proposals for standardisation.[7]

BaseNameUsage
2BinaryDigital computing
3TernaryCantor 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
4QuaternaryData transmission, DNA bases and Hilbert curves; Chumashan languages, and Kharosthi numerals
5QuinaryGumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6SenaryDiceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7Septenaryweeks timekeeping
8OctalCharles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian (I Ching, China)
9NonaryBase9 encoding
10DecimalMost widely used by modern civilizations[8][9][10]
11UndecimalJokingly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal; check digits in ISBN
12DuodecimalLanguages 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
13TridecimalConway base 13 function
14TetradecimalProgramming for the HP 9100A/B calculator[11] and image processing applications[12]; pound and stone
15PentadecimalTelephony routing over IP, and the Huli language
16HexadecimalBase16 encoding; compact notation for binary data; tonal system; ounce and pound
17HeptadecimalBase17 encoding
18OctodecimalBase18 encoding
20VigesimalBasque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
23TrivigesimalKalam language, Kobon language
24Tetravigesimal24-hour clock timekeeping; Kaugel language
26HexavigesimalBase 26 encoding; sometimes used for encryption or ciphering.[13]
27HeptavigesimalTelefol and Oksapmin 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,[14] to provide a concise encoding of alphabetic strings,[15] or as the basis for a form of gematria.[16]
30TrigesimalThe Natural Area Code
32DuotrigesimalBase32 encoding and the Ngiti language
33TritrigesimalUse of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong
36HexatrigesimalBase36 encoding; use of letters with digits
40QuadragesimalDEC Radix-50₈ 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.
52DuoquinquagesimalBase52 encoding, a variant of Base62 without vowels[17]
56HexaquinquagesimalBase56 encoding, a variant of Base58[18]
57HeptaquinquagesimalBase57 encoding, a variant of Base62 excluding I, O, l, U, and u[19]
58OctoquinquagesimalBase58 encoding
60SexagesimalBabylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[20] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
62DuosexagesimalBase62 encoding, using 0–9, A–Z, and a–z
64TetrasexagesimalBase64 encoding; I Ching in China.
This system is conveniently coded into ASCII by using the 26 letters of the Latin alphabet in both upper and lower case (52 total) plus 10 numerals (62 total) and then adding two special characters (for example, YouTube video codes use the hyphen and underscore characters, - and _ to total 64).
85PentoctogesimalAscii85 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.
91UnnonagesimalBase91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92DuononagesimalBase92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[21]
93TrinonagesimalBase93 encoding, using all of ASCII printable characters except for "," (0x27) and "-" (0x3D) as well as the Space character. "," is reserved for delimiter and "-" is reserved for negation.[22]
94TetranonagesimalBase94 encoding, using all of ASCII printable characters.[23]
95PentanonagesimalBase95 encoding, a variant of Base94 with the addition of the Space character.[24]

Non-standard positional numeral systems

Bijective numeration

BaseNameUsage
1Unary (Bijective base-1)Tally marks
10Bijective base-10
26Bijective base-26Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.[25]

Signed-digit representation

BaseNameUsage
2Balanced binary (Non-adjacent form)
3Balanced ternaryTernary computers
5Balanced quinary
9Balanced nonary
10Balanced decimalJohn Colson
Augustin Cauchy

Negative bases

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

BaseNameUsage
−2Negabinary
−3Negaternary
−10Negadecimal

Complex bases

BaseNameUsage
2iQuater-imaginary base
−1 ± iTwindragon baseTwindragon fractal shape

Non-integer bases

BaseNameUsage
φGolden ratio baseEarly Beta encoder[26]
eBase Lowest radix economy

Other

Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[27], as are many developed later, such as the Roman numerals.

See also

References

  1. ↑ https://books.google.nl/books?id=kXZhBAAAQBAJ&pg=PA148&dq=ethiopic+numerals+coptic&hl=nl&sa=X&redir_esc=y#v=onepage&q=ethiopic&f=false
  2. ↑ Everson, Michael (2007-07-25). "Proposal to add two numbers for the Phoenician script" (PDF). UTC Document Register. L2/07-206 (WG2 N3284): Unicode Consortium.
  3. ↑ Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
  4. ↑ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
  5. ↑ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 2017.
  6. ↑ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669 .
  7. ↑ http://www.numberbases.com/terms/BaseNames.pdf
  8. ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  9. ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
  10. ↑ 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
  11. ↑ HP 9100A/B programming, HP Museum
  12. ↑ Free Patents Online
  13. ↑ http://www.dcode.fr/base-26-cipher
  14. ↑ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proceedings. AMIA Symposium: 305–309, PMC 2244404, PMID 12463836 .
  15. ↑ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183 .
  16. ↑ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77 .
  17. ↑ "Base52". Retrieved 2016-01-03.
  18. ↑ "Base56". Retrieved 2016-01-03.
  19. ↑ "Base57". Retrieved 2016-01-03.
  20. ↑ "NewBase60". Retrieved 2016-01-03.
  21. ↑ "Base92". Retrieved 2016-01-03.
  22. ↑ "Base93". Retrieved 2017-02-13.
  23. ↑ "Base94". Retrieved 2016-01-03.
  24. ↑ "base95 Numeric System". Retrieved 2016-01-03.
  25. ↑ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
  26. ↑ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324&ndash, 4334, arXiv:0806.1083, doi:10.1109/TIT.2008.928235
  27. ↑ 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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