List of numeral systems

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

This is a list of numeral systems.

Positional notation

These numeral systems use place-value notation.

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.^{[citation needed]}

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-tablepoon-shot measurement systems; most economical integer base system
4	Quaternary	Data transmission and Hilbert curves; Chumashan languages, Ventureño language, and Kharosthi numerals
5	Quinary	Gumatj language, Nunggubuyu language, Kuurn Kopan Noot language, and Saraveca
6	Senary	Diceware, Ndom language, Proto-Uralic (suspected)
7	Septenary	Week cycle
8	Octal	Charles XII of Sweden, Unix-like permissions, DEC PDP-11
9	Nonary
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 such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara
13	Tridecimal	A cycle of the Maya calendar
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
18	Octodecimal	A cycle of the Mesoamerican Long Count calendar
20	Vigesimal	Celtic numerals, Maya numerals, Yoruba numerals, Tlingit people, Dzongkha numerals, Santali language, Ainu language
24	Tetravigesimal	Umbu-Ungu also known as Kakoli.
25	Pentavigesimal	Compact notation of quinary numbers
26	Hexavigesimal
27	Septemvigesimal	Telefol and Oksapmin languages, and compact notation of ternary numbers
30	Trigesimal	Month cycle for various calendars
32	Duotrigesimal	Base32 encoding, and the Ngiti language
36	Hexatrigesimal	Base36 encoding, and compact notation of senary numbers
60	Sexagesimal	Babylonian numeral system; degrees-minutes-seconds and hours-minutes-seconds measurement systems
64	Tetrasexagesimal	Base64 encoding; compact notation of quaternary or of octal numbers

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
2	Non-adjacent form
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
e	Base $e$	Lowest radix economy
π	Base $\pi$
√2	Base $\sqrt{2}$
¹²√2	Base $\sqrt[12]{2}$	Scientific pitch notation

Other

• Mixed radix
• Quote notation

Non-positional notation

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

Bijective unary numeration

Base	Name	Usage
1	Unary	Tally marks

Numerals

Name	Base	Sample	Approx. first appearance
Babylonian numerals	60		3100 BC
Greek numerals	10	α β γ δ ε ϝ ζ η θ ι	After 100 BC
Roman numerals	10	I II III IV V VI VII VIII IX X	1000 BC
Chinese rod numerals	10		1st century
Arabic numerals	10	0 1 2 3 4 5 6 7 8 9 10	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

See also

• Radix
• Radix economy
• List of numbers in various languages