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{{Numeral systems}}
{{Expand list|date=March 2011}}

This is a '''list of [[numeral system]]s'''.

==[[Positional notation]]==
These numeral systems use place-value notation.

===Standard positional numeral systems===
[[File:Binary clock.svg|thumb|A [[binary clock]] might use [[Light-emitting diode|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 [[Hexadecimal#Etymology|somewhat arbitrarily]] from a mix of [[Latin]] and [[Greek language|Greek]].{{citation needed|date=March 2013}}

{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2 || [[Binary number|Binary]] || Digital [[computing]]
|-
| 3 || [[Ternary numeral system|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 ''[[radix economy|economical]]'' integer base
|-
| 4 || [[Quaternary numeral system|Quaternary]] || Data transmission and [[Hilbert curve]]s; [[Chumashan languages]], and [[Kharosthi#Numerals|Kharosthi numerals]]
|-
| 5 || [[Quinary]] || [[Gumatj language|Gumatj]], [[Nunggubuyu language|Nunggubuyu]], [[Kuurn Kopan Noot language|Kuurn Kopan Noot]], and [[Saraveca language|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#Scientific contributions|Charles XII of Sweden]], [[Filesystem_permissions#Octal_notation|Unix-like permissions]], [[PDP-11_architecture#Memory_management|DEC PDP-11]], compact notation for binary numbers
|-
| 9 || [[Nonary]] || compact notation of ternary numbers
|-
| 10 || [[Decimal]] || Most widely used by modern civilizations<ref>''The History of Arithmetic'', [[Louis Charles Karpinski]], 200pp, Rand McNally & Company, 1925.</ref><ref>''Histoire universelle des chiffres'', [[Georges Ifrah]], Robert Laffont, 1994.</ref><ref>''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</ref>
|-
| 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 [[Nigeria]]n Middle Belt [[Janji language|Janji]], [[Gbiri-Niragu language|Gbiri-Niragu]], [[Piti language|Piti]], and the Nimbia dialect of [[Gwandara language|Gwandara]]; [[Chepang language]] of [[Nepal]], and the Mahl dialect of [[Maldivian language|Maldivian]]; dozen-gross-great gross counting; hours and months timekeeping; years of [[Chinese zodiac]]
|-
| 13 || [[Base 13|Tridecimal]] || A cycle of the [[Maya calendar]]
|-
| 14 || [[Tetradecimal]] || Programming for the [[Hewlett-Packard|HP]] 9100A/B calculator<ref>[http://www.hpmuseum.org/prog/hp9100pr.htm HP Museum]</ref> and image processing applications<ref>[http://www.freepatentsonline.com/6690378.html Free Patents Online]</ref>
|-
| 15 || [[Pentadecimal]] || Telephony routing over IP, and the [[Huli language]]
|-
| 16 || [[Hexadecimal]] || [[Base16]] encoding; compact notation for [[binary data]] or quaternary numbers
|-
| 18 || Octodecimal || A cycle of the [[Mesoamerican Long Count calendar]]
|-
| 20 || [[Vigesimal]] || [[Yan Tan Tethera|Celtic]], [[Maya numerals|Maya]], [[Inuit numerals|Inuit]], [[Yoruba numerals|Yoruba]], [[Tlingit people|Tlingit]], and [[Dzongkha numerals|Dzongkha]] numerals; [[Santali language|Santali]], and [[Ainu language|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 language|Telefol]] and [[Oksapmin language|Oksapmin]] languages; compact notation of ternary numbers
|-
| 28 || Octovigesimal || Four week month of thirteen month calendar
|-
| 30 || Trigesimal || [[Month#Months in various calendars|Month cycle]] for various calendars; most efficiently limits recurring digits for fractions; 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
|-
| 64 || [[Tetrasexagesimal]] || [[Base64]] encoding; compact notation of binary, quaternary or octal numbers
|-
| 85 || Pentaoctagesimal || [[Ascii85]] encoding
|-
| 120 || Centovigesimal || [[Great hundred]]
|-
| 240 || Duocentoquadragesimal || [[£sd]]
|-
| 360 || Trecentosexagesimal || [[Degree (angle)|Degree]] division of circle
|}

===[[Non-standard positional numeral systems]]===

====[[Bijective numeration]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 1 || [[Unary numeral system]], used in [[tally mark|tallying]] ||
|-
| 10 || [[Bijective numeration#The_bijective_base-10_system|Bijective base-10]] ||
|-
| 26 || [[Hexavigesimal#Bijective base 26|Bijective base-26]] || Spreadsheet column numeration
|}

====[[Signed-digit representation]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2 || [[Non-adjacent form]] ||
|-
| 3 || [[Balanced ternary]] || [[Ternary computer]]s
|-
|10 || Balanced decimal || [[John Colson]]<br>[[Augustin Cauchy]]
|}

====[[Negative bases]]====
The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:

{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| −2 || Negabinary ||
|-
| −3 || Negaternary ||
|}

====[[Complex base systems|Complex bases]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 2''i'' || [[Quater-imaginary base]] ||
|-
| −1 ± ''i'' || [[Twindragon]] base || [[Twindragon]] fractal shape
|}

====[[Non-integer representation|Non-integer bases]]====
{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| [[Golden ratio|φ]] || [[Golden ratio base]] || Early [[Beta encoder]]<ref>
{{Citation
|last= Ward |first=Rachel
|year=2008
|title= On Robustness Properties of Beta Encoders and Golden Ratio Encoders
|journal=[[IEEE Transactions on Information Theory]]
|volume=54 |issue=9 |pages= 4324–4334
|doi=10.1109/TIT.2008.928235
}}</ref>
|-
| [[e (mathematical constant)|e]] || Base <math>e</math> || Lowest [[radix economy]]
|-
| [[π]] || Base <math>\pi</math> ||
|-
| [[√2]] || Base <math>\sqrt{2}</math> ||
|-
| [[Twelfth root of two|¹²√2]] || Base <math>\sqrt[12]{2}</math> || [[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===

{| class="wikitable"
|-
! Base !! Name !! Usage
|-
| 1 || [[Unary numeral system|Unary]] || [[Tally marks]]
|}

==Numerals==
{| class="wikitable sortable"
|-
! Name !! Base !! Sample !! Approx. first appearance
|-
| [[Babylonian numerals]] || 60 || [[File:Babylonian 1.svg|15px]][[File:Babylonian 2.svg|15px]][[File:Babylonian 3.svg|15px]][[File:Babylonian 4.svg|15px]][[File:Babylonian 5.svg|15px]][[File:Babylonian 6.svg|15px]][[File:Babylonian 7.svg|15px]][[File:Babylonian 8.svg|15px]][[File:Babylonian 9.svg|15px]][[File:Babylonian 10.svg|15px]] || 3100 BC
|-
| [[Egyptian numerals]] || 10 || <hiero>Z1</hiero> <hiero>V20</hiero> <hiero>V1</hiero> <hiero>M12</hiero> <hiero>D50</hiero> <hiero>I8</hiero> <br>or<br><hiero>I7</hiero> <hiero>C11</hiero> || 3000 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 || [[File:Counting rod v1.png|x15px]] [[File:Counting rod v2.png|x15px]] [[File:Counting rod v3.png|x15px]] [[File:Counting rod v4.png|x15px]] [[File:Counting rod v5.png|x15px]] [[File:Counting rod v6.png|x15px]] [[File:Counting rod v7.png|x15px]] [[File:Counting rod v8.png|x15px]] [[File:Counting rod v9.png|x15px]] [[File:Counting rod h1.png|15px]] || 1st century
|-
| [[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
|}

==See also==
* [[Radix]]
* [[Radix economy]]
* [[List of numbers in various languages]]
* [[Numeral prefix]]

==References==
<references/>

[[Category:Numeral systems]]

Self-Sampling Assumption - Revision history

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - SIA