# List of numeral systems

This is a list of numeral systems.

## By culture

Name Base Sample Approx. first appearance
Babylonian numerals 60 3100 BC
Egyptian numerals 10

or
3000 BC
Maya numerals 20
Oracle bone script 0 0 14th century BC?
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) 10 零 一 二 三 四 五 六 七 八 九
Roman numerals 10 Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ 1000 BC
Greek numerals 10 α β γ δ ε ϝ ζ η θ ι After 100 BC
Chinese rod numerals 10 1st century
Hindu-Arabic Numerals 10 0 1 2 3 4 5 6 7 8 9 9th century
John Napier's Location arithmetic 2 a b ab c ac bc abc d ad bd 1617 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.[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 language, and Proto-Uralic language (suspected)
7 Septenary Week cycle
8 Octal Charles XII of Sweden, Unix-like permissions, DEC PDP-11, compact notation for binary numbers
9 Nonary compact notation of ternary numbers
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; hours and months timekeeping; years of Chinese zodiac; foot and inch.
13 Tridecimal A cycle of the Maya calendar
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 or quaternary numbers; tonal system
18 Octodecimal A cycle of the Mesoamerican Long Count calendar
20 Vigesimal Celtic, Maya, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
24 Tetravigesimal Kaugel language; hours timekeeping
25 Pentavigesimal Compact notation of quinary numbers
26 Hexavigesimal Uses of letters without digits, e.g. spreadsheet column numeration
27 Septemvigesimal Telefol and Oksapmin languages; compact notation of ternary numbers
28 Octovigesimal Four week month of thirteen month calendar
30 Trigesimal Month cycle for various calendars; The Natural Area Code
32 Duotrigesimal Base32 encoding; compact notation of binary; and the Ngiti language
36 Hexatrigesimal Base36 encoding; use of letters with digits; compact notation of senary numbers
60 Sexagesimal Babylonian numerals; degrees-minutes-seconds and hours-minutes-seconds measurement systems
62 Duosexagesimal Base62 encoding; using all English letters (capital and lowercase) and digits but no others, e.g. URL shortening.
64 Tetrasexagesimal Base64 encoding; compact notation of binary, quaternary or octal numbers
85 Pentaoctagesimal Ascii85 encoding
120 Centovigesimal Great hundred
256 Internally in computers
360 Trecentosexagesimal Degree division of circle

### Non-standard positional numeral systems

#### Bijective numeration

Base Name Usage
10 Bijective base-10
26 Bijective base-26 Spreadsheet column numeration

#### Signed-digit representation

Base Name Usage
3 Balanced ternary Ternary computers
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[7]
e Base $e$ Lowest radix economy
π Base $\pi$
√2 Base $\sqrt{2}$
¹²√2 Base $\sqrt[12]{2}$ Scientific pitch notation

### Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional.[8]

Base Name Usage
1 Unary Tally marks