List of numeral systems


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

By culture / time period

By type of notation

Numeral systems are classified here as to whether they use positional notation, and further categorized by radix or base.

Standard positional numeral systems

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. There have been some proposals for standardisation.
BaseNameUsage
2BinaryDigital computing, imperial and customary volume
3TernaryCantor set ; 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
7SeptenaryWeeks timekeeping, Western music letter notation
8OctalCharles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian
9NonaryBase9 encoding; compact notation for ternary
10Decimal / DenaryMost widely used by modern civilizations
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 digit in ISBN-10. A base-11 number system was attributed to the Māori in the 19th century and the Pangwa in the 20th century.
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
13TridecimalBase13 encoding; Conway base 13 function
14TetradecimalProgramming for the HP 9100A/B calculator and image processing applications; 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
19EnneadecimalBase19 encoding
20VigesimalBasque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
21UnvigesimalBase21 encoding
22DuovigesimalBase22 encoding
23TrivigesimalKalam language, Kobon language
24Tetravigesimal24-hour clock timekeeping; Kaugel language
25PentavigesimalBase25 encoding
26HexavigesimalBase26 encoding; sometimes used for encryption or ciphering, using all letters
27Heptavigesimal SeptemvigesimalTelefol 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, to provide a concise encoding of alphabetic strings, or as the basis for a form of gematria. Compact notation for ternary.
28OctovigesimalBase28 encoding; months timekeeping
29EnneavigesimalBase29
30TrigesimalThe 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
31UntrigesimalBase31
32DuotrigesimalBase32 encoding and the Ngiti language
33TritrigesimalUse of letters with digits in vehicle registration plates of Hong Kong
34TetratrigesimalUsing all numbers and all letters except I and O
35PentatrigesimalUsing all numbers and all letters except O
36HexatrigesimalBase36 encoding; use of letters with digits
37HeptatrigesimalBase37; using all numbers and all letters of the Spanish alphabet
38OctotrigesimalBase38 encoding; use all duodecimal digits and all letters
40QuadragesimalDEC 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.
42DuoquadragesimalBase42 encoding
45PentaquadragesimalBase45 encoding
48OctoquadragesimalBase48 encoding
49EnneaquadragesimalCompact notation for septenary
50QuinquagesimalBase50 encoding; SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers.
52DuoquinquagesimalBase52 encoding, a variant of Base62 without vowels or a variant of Base26 using all lower and upper case letters.
54TetraquinquagesimalBase54 encoding
56HexaquinquagesimalBase56 encoding, a variant of Base58
57HeptaquinquagesimalBase57 encoding, a variant of Base62 excluding I, O, l, U, and u or I, 1, l, 0, and O
58OctoquinquagesimalBase58 encoding
60SexagesimalBabylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _; degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
62Duosexagesimal 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 plus 10 numerals and then adding two special characters.
72DuoseptuagesimalBase72 encoding
80OctogesimalBase80 encoding
81UnoctogesimalBase81 encoding, using as 81=34 is related to ternary
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.
90NonagesimalRelated to Goormaghtigh conjecture for the generalized repunit numbers.
91UnnonagesimalBase91 encoding, using all ASCII except "-", "\", and "'" ; one variant uses "\" in place of """.
92DuononagesimalBase92 encoding, using all of ASCII except for "`" and """ due to confusability.
93TrinonagesimalBase93 encoding, using all of ASCII printable characters except for "," and "-" as well as the Space character. "," is reserved for delimiter and "-" is reserved for negation.
94TetranonagesimalBase94 encoding, using all of ASCII printable characters.
95PentanonagesimalBase95 encoding, a variant of Base94 with the addition of the Space character.
96HexanonagesimalBase96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits
100CentesimalAs 100=102, these are two decimal digits
120CentevigesimalBase120 encoding
121CenteunvigesimalRelated to base 11
125CentepentavigesimalRelated to base 5
128CenteoctovigesimalUsing as 128=27
144CentetetraquadragesimalTwo duodecimal digits
256DuocentehexaquinquagesimalBase256 encoding, as 256=28
360TrecentosexagesimalDegrees for angle

[Non-standard positional numeral systems]

[Bijective numeration]

[Signed-digit representation]

[Negative bases]

The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:
BaseNameUsage
−2Negabinary
−3Negaternary
−4Negaquaternary
−5Negaquinary
−6Negasenary
−8Negaoctal
−10Negadecimal
−12Negaduodecimal
−16Negahexadecimal

Complex bases">Complex-base system">Complex bases

Non-integer bases">Non-integer representation">Non-integer bases

n-adic number">p-adic number">n-adic number

BaseNameUsage
2Dyadic number
3Triadic number
4Tetradic numberthe same as dyadic number
5Pentadic number
6Hexadic numbernot a field
7Heptadic number
8Octadic numberthe same as dyadic number
9Enneadic numberthe same as triadic number
10Decadic numbernot a field
11Hendecadic number
12Dodecadic numbernot a field

[Mixed radix]

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