# List of numeral systems

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
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 <15th century
Muisca numerals 20 <15th century
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) 10 〇/零 一 二 三 四 五 六 七 八 九 十
Roman numerals 10 N
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ
L C D M
1000 BC
Hebrew numerals 10 א ב ג ד ה ו ז ח ט
י כ ל מ נ ס ע פ צ
800 BC
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
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
Chinese rod numerals 10 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 1st century
Eastern Arabic numerals 10 ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ 8th century
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
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
Burmese numerals 10 ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉

## 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)
7 Septenary weeks timekeeping
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; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Telephone number
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; ounce and pound
20 Vigesimal Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
23 Trivigesimal Kalam language, Kobon language
24 Tetravigesimal 24-hour clock timekeeping; Kaugel language
26 Hexavigesimal Base 26 encoding; sometimes used for encryption or ciphering.[7]
27 Heptavigesimal Telefol 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,[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
33 Tritrigesimal Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong
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]
93 Trinonagesimal Base93 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.[18]
94 Tetranonagesimal Base94 encoding, using all of ASCII printable characters.[19]
95 Pentanonagesimal Base95 encoding, a variant of Base94 with the addition of the Space character.[20]
256 Ducentahexaquinquagesimal Base256 encoding[citation needed]

### 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.[21]

#### Signed-digit representation

Base Name Usage
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

#### 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[22]
e Base $e$ Lowest radix economy

### Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional.[23]

## Trivia

• 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.

## 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
9. Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
10. Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
11. "Base52". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
12. "Base56". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
13. "Base57". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
14. "NewBase60". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
15. "NewBase61". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
16. "Base65 Encoding". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
17. "Base92". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
18. "Base93". Retrieved 2017-02-13.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
19. "Base94". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
20. "base95 Numeric System". Retrieved 2016-01-03.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
21. Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
22. 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
23. 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.