# List of numeral systems

(Redirected from Undenary)

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
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)
8 Octal Charles XII of Sweden, Unix-like permissions, DEC PDP-11, compact notation for binary 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 Conway base 13 function
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; tonal system
20 Vigesimal Celtic, Maya, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
24 Tetravigesimal Kaugel language
26 Hexavigesimal Uses of letters without digits, e.g. spreadsheet column numeration
27 Septemvigesimal Telefol and Oksapmin languages
30 Trigesimal The Natural Area Code
32 Duotrigesimal Base32 encoding and the Ngiti language
36 Hexatrigesimal Base36 encoding; use of letters with digits
40 Quadragesimal Known in Paya (as of 1928) and Kashaya; was used in Hawaiian, but later became decimal under foreign influence.
60 Sexagesimal Babylonian numerals; degrees-minutes-seconds and hours-minutes-seconds measurement systems; known in Ekari and Ntomba (later became decimal), as well as the extinct Sumerian language
64 Tetrasexagesimal Base64 encoding
80 Octogesimal Known in a few West African languages
85 Pentoctogesimal Ascii85 encoding

### 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 $\pi$ "Pi-nary"
√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