List of numeral systems

See also: List of types of numbers

This list is incomplete; you can help by expanding it.

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.

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
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 and image processing applications
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.
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, to provide a concise encoding of alphabetic strings, or as the basis for a form of gematria.
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
56	Hexaquinquagesimal	Base56 encoding, a variant of Base58
57	Heptaquinquagesimal	Base57 encoding, a variant of Base62 excluding I, O, l, U, and u
58	Octoquinquagesimal	Base58 encoding
60	Sexagesimal	Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
61	Unsexagesimal	NewBase61 encoding, variant of NewBase60 with a space
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
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.
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.
94	Tetranonagesimal	Base94 encoding, using all of ASCII printable characters.
95	Pentanonagesimal	Base95 encoding, a variant of Base94 with the addition of the Space character.
256	Ducentahexaquinquagesimal	Base256 encoding

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.

Signed-digit representation

Base	Name	Usage
2	Balanced binary (Non-adjacent form)
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
−10	Negadecimal

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
e	Base $e$	Lowest radix economy

Other

• Mixed radix
• Quote notation
• Redundant binary representation
• hereditary base-n notation

Non-positional notation

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

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.

See also

• Radix
• Radix economy
• Table of bases
• List of numbers in various languages (cardinal number names)
• Numeral prefix