List of numeral systems

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

By culture / time period

Name

Base

Sample

Approx. First Appearance

Prehistoric numerals

35,000 BCE

Babylonian numerals

60

3,100 BCE

Egyptian numerals

10

3,000 BCE

Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean), Vietnamese numerals (Sino-Vietnamese)

10

零一二三四五六七八九十百千萬億 (Default, Traditional Chinese)

〇一二三四五六七八九十百千万亿 (Default, Simplified Chinese)

零壹貳叄肆伍陸柒捌玖拾佰仟萬億 (Financial, T. Chinese)

零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (Financial, S. Chinese)

1,600 BCE

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 (

)

1,500 BCE

Roman numerals

10

I V X L C D M

1,000 BCE

Hebrew numerals

10

א ב ג ד ה ו ז ח ט
י כ ל מ נ ס ע פ צ ק ר ש ת ך ם ף ץ

800 BCE

Indian numerals

10

Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९ Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩

750 – 690 BCE

Greek numerals

10

ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ

<400 BCE

Phoenician numerals

10

𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [1]

<250 BCE[2]

Chinese rod numerals

10

𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩

1st Century

Ge'ez numerals

10

፩, ፪, ፫, ፬, ፭, ፮, ፯, ፰, ፱
፲, ፳, ፴, ፵, ፶, ፷, ፸, ፹, ፺, ፻

3rd – 4th Century, 15th Century (Modern Style)[3]

Armenian numerals

10

Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ

Early 5th Century

Khmer numerals

10

០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩

Early 7th Century

Thai numerals

10

๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙

7th Century[4]

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

<8th 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

Cyrillic numerals

10

А҃ В҃ Г҃ Д҃ Е҃ Ѕ҃ З҃ И҃ Ѳ҃ І҃ ...

10th Century

Tangut numerals

10

𘈩𗍫𘕕𗥃𗏁𗤁𗒹𘉋𗢭𗰗

1,036 CE

Burmese numerals

10

၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉

11th Century[5]

Maya numerals

20

<15th Century

Muisca numerals

20

<15th Century

Aztec numerals

20

16th Century

Sinhala numerals

10

෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯

𑇡 𑇢 𑇣 𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴

<18th Century

Kaktovik Inupiaq numerals

20

1994

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]

Base	Name	Usage
2	Binary	Digital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon)
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, DNA bases and Hilbert curves; Chumashan languages, and Kharosthi numerals
5	Quinary	Gumatj, Ateso, 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)
9	Nonary	Base9 encoding; compact notation for ternary
10	Decimal / Denary	Most widely used by modern civilizations[8][9][10]
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; check digit in ISBN-10
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; Roman fractions
13	Tridecimal	Base13 encoding; Conway base 13 function
14	Tetradecimal	Programming for the HP 9100A/B calculator[11] and image processing applications;[12] pound and stone
15	Pentadecimal	Telephony routing over IP, and the Huli language
16	Hexadecimal	Base16 encoding; compact notation for binary data; tonal system; ounce and pound
17	Heptadecimal	Base17 encoding
18	Octodecimal	Base18 encoding
19	Enneadecimal	Base19 encoding
20	Vigesimal	Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
21	Unvigesimal	Base21 encoding
22	Duovigesimal	Base22 encoding
23	Trivigesimal	Kalam language, Kobon language
24	Tetravigesimal	24-hour clock timekeeping; Kaugel language
25	Pentavigesimal	Base25 encoding
26	Hexavigesimal	Base26 encoding; sometimes used for encryption or ciphering,[13] using all letters
27	Heptavigesimal Septemvigesimal	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,[14] to provide a concise encoding of alphabetic strings,[15] or as the basis for a form of gematria.[16] Compact notation for ternary.
28	Octovigesimal	Base28 encoding; months timekeeping
29	Enneavigesimal	Base29
30	Trigesimal	The Natural Area Code, this is the smallest base such that all of 1/2 to 1/6 terminate, a number n is a regular number if and only if 1/n terminates in base 30
31	Untrigesimal	Base31
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
34	Tetratrigesimal	Using all numbers and all letters except I and O
35	Pentatrigesimal	Using all numbers and all letters except O
36	Hexatrigesimal	Base36 encoding; use of letters with digits
37	Heptatrigesimal	Base37; using all numbers and all letters of the Spanish alphabet
38	Octotrigesimal	Base38 encoding; use all duodecimal digits and all letters
40	Quadragesimal	DEC RADIX 50/MOD40 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.
42	Duoquadragesimal	Base42 encoding
45	Pentaquadragesimal	Base45 encoding
48	Octoquadragesimal	Base48 encoding
49	Enneaquadragesimal	Compact notation for septenary
50	Quinquagesimal	Base50 encoding; SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers.
52	Duoquinquagesimal	Base52 encoding, a variant of Base62 without vowels[17]
54	Tetraquinquagesimal	Base54 encoding
56	Hexaquinquagesimal	Base56 encoding, a variant of Base58[18]
57	Heptaquinquagesimal	Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[19] or I, 1, l, 0, and O [20]
58	Octoquinquagesimal	Base58 encoding
60	Sexagesimal	Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[21] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
62	Duosexagesimal	Base62 encoding, using 0–9, A–Z, and a–z
64	Tetrasexagesimal	Base64 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).
72	Duoseptuagesimal	Base72 encoding
80	Octogesimal	Base80 encoding
81	Unoctogesimal	Base81 encoding, using as 81=3⁴ is related to ternary
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.
90	Nonagesimal	Related to Goormaghtigh conjecture for the generalized repunit numbers.
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.[22]
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.[23]
94	Tetranonagesimal	Base94 encoding, using all of ASCII printable characters.[24]
95	Pentanonagesimal	Base95 encoding, a variant of Base94 with the addition of the Space character.[25]
96	Hexanonagesimal	Base96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits
100	Centesimal	As 100=10², these are two decimal digits
120	Centevigesimal	Base120 encoding
121	Centeunvigesimal	Related to base 11
125	Centepentavigesimal	Related to base 5
128	Centeoctovigesimal	Using as 128=2⁷
144	Centetetraquadragesimal	Two duodecimal digits
256	Duocentehexaquinquagesimal	Base256 encoding, as 256=2⁸
360	Trecentosexagesimal	Degrees for angle

Non-standard positional numeral systems

Bijective numeration

Base	Name	Usage
1	Unary (Bijective base-1)	Tally marks
2	Bijective base-2
3	Bijective base-3
4	Bijective base-4
5	Bijective base-5
6	Bijective base-6
8	Bijective base-8
10	Bijective base-10
12	Bijective base-12
16	Bijective base-16
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.[26]

Signed-digit representation

Base	Name	Usage
2	Balanced binary (Non-adjacent form)
3	Balanced ternary	Ternary computers
4	Balanced quaternary
5	Balanced quinary
6	Balanced senary
7	Balanced septenary
8	Balanced octal
9	Balanced nonary
10	Balanced decimal	John Colson Augustin Cauchy
11	Balanced undecimal
12	Balanced duodecimal

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
−4	Negaquaternary
−5	Negaquinary
−6	Negasenary
−8	Negaoctal
−10	Negadecimal
−12	Negaduodecimal
−16	Negahexadecimal

Complex bases

Base	Name	Usage
2i	Quater-imaginary base	related to base −4 and base 16
${\sqrt {2}}i$	Base ${\sqrt {2}}i$	related to base −2 and base 4
${\sqrt[{4}]{2}}i$	Base ${\sqrt[{4}]{2}}i$	related to base 2
$2\omega$	Base $2\omega$	related to base 8
${\sqrt[{3}]{2}}\omega$	Base ${\sqrt[{3}]{2}}\omega$	related to base 2
−1 ± i	Twindragon base	Twindragon fractal shape, related to base −4 and base 16
1 ± i	Nega-Twindragon base	related to base −4 and base 16

Non-integer bases

Base	Name	Usage
${\frac {3}{2}}$	Base ${\frac {3}{2}}$	a rational non-integer base
${\frac {4}{3}}$	Base ${\frac {4}{3}}$	related to duodecimal
${\frac {5}{2}}$	Base ${\frac {5}{2}}$	related to decimal
${\sqrt {2}}$	Base ${\sqrt {2}}$	related to base 2
${\sqrt {3}}$	Base ${\sqrt {3}}$	related to base 3
${\sqrt[{3}]{2}}$	Base ${\sqrt[{3}]{2}}$
${\sqrt[{4}]{2}}$	Base ${\sqrt[{4}]{2}}$
${\sqrt[{12}]{2}}$	Base ${\sqrt[{12}]{2}}$	using in music scale
$2{\sqrt {2}}$	Base $2{\sqrt {2}}$
$-{\frac {3}{2}}$	Base $-{\frac {3}{2}}$	a negative rational non-integer base
$-{\sqrt {2}}$	Base $-{\sqrt {2}}$	a negative non-integer base, related to base 2
${\sqrt {10}}$	Base ${\sqrt {10}}$	related to decimal
$2{\sqrt {3}}$	Base $2{\sqrt {3}}$	related to duodecimal
φ	Golden ratio base	Early Beta encoder[27]
ρ	Plastic number base
ψ	Supergolden ratio base
$1+{\sqrt {2}}$	Silver ratio base
e	Base $e$	Lowest radix economy
π	Base $\pi$
$e^{\pi }$	Base $e^{\pi }$

n-adic number

Base	Name	Usage
2	Dyadic number
3	Triadic number
4	Tetradic number	the same as dyadic number
5	Pentadic number
6	Hexadic number	not a field
7	Heptadic number
8	Octadic number	the same as dyadic number
9	Enneadic number	the same as triadic number
10	Decadic number	not a field
11	Hendecadic number
12	Dodecadic number	not a field

Mixed radix

Factorial number system {1, 2, 3, 4, 5, 6, ...}
Even double factorial number system {2, 4, 6, 8, 10, 12, ...}
Odd double factorial number system {1, 3, 5, 7, 9, 11, ...}
Primorial number system {2, 3, 5, 7, 11, 13, ...}
{60, 60, 24, 7} in timekeeping
{60, 60, 24, 30 (or 31 or 28 or 29), 12} in timekeeping
(12, 20) traditional English monetary system (£sd)
(20, 18, 13) Maya timekeeping

Other

Quote notation
Redundant binary representation
Hereditary base-n notation
Asymmetric numeral systems optimized for non-uniform probability distribution of symbols

Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[28] as are many developed later, such as the Roman numerals.

References

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.CS1 maint: location (link)
Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
Chrisomalis, Stephen (2010-01-18). Numerical Notation: A Comparative History. ISBN 9781139485333.
Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
"Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 2017.
For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
http://www.numberbases.com/terms/BaseNames.pdf
The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
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
HP 9100A/B programming, HP Museum
Free Patents Online
http://www.dcode.fr/base-26-cipher
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.
Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
"Base52". Retrieved 2016-01-03.
"Base56". Retrieved 2016-01-03.
"Base57". Retrieved 2016-01-03.
"Base57". Retrieved 2019-01-22.
"NewBase60". Retrieved 2016-01-03.
"Base92". Retrieved 2016-01-03.
"Base93". Retrieved 2017-02-13.
"Base94". Retrieved 2016-01-03.
"base95 Numeric System". Retrieved 2016-01-03.
Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, arXiv:0806.1083, Bibcode:2008arXiv0806.1083W, doi:10.1109/TIT.2008.928235
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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[1] 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.CS1 maint: location (link)

[Cajori-2] Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.

[3] Chrisomalis, Stephen (2010-01-18). Numerical Notation: A Comparative History. ISBN 9781139485333.

[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] ttp://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] ttp://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] "Base57". Retrieved 2019-01-22.

[21] "NewBase60". Retrieved 2016-01-03.

[22] "Base92". Retrieved 2016-01-03.

[23] "Base93". Retrieved 2017-02-13.

[24] "Base94". Retrieved 2016-01-03.

[25] "base95 Numeric System". Retrieved 2016-01-03.

[26] Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.

[27] Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, arXiv:0806.1083, Bibcode:2008arXiv0806.1083W, doi:10.1109/TIT.2008.928235

[28] 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.