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# Sexagesimal numeral system

(Redirected from Sexagesimal)

The sexagesimal numeral system[1] is a place-value notation using the powers of 60 (see A159991) rather than the powers of 10. Sexagesimal was extensively used by the ancient Babylonians, albeit without a zero (they used extra spacing instead). Although the decimal numeral system has effectively eliminated all competition for human use, like duodecimal, sexagesimal shows many vestiges in our current usage, like the subdivisions of time (an hour into 60 minutes, and a minute into 60 seconds) or angle (an arc degree into 60 arc minutes, and a arc minute into 60 arc seconds).

The Babylonians used cuneiform number characters, and although Unicode does acknowledge cuneiform characters, their use on computers remains rather impractical. Some computer algebra systems like Wolfram Mathematica support bases from base 2 up to base 36 by the use of the alphabetic characters A to Z for the digits 11 to 35; sexagesimal of course also needs digits 36 to 59.

A solution sometimes used is to differentiate between upper and lower case, so that "A" is 10 but "a" is 36. Thus, for example, 1729 (1001 in duodecimal, 6C1 in hexadecimal) is Sn in sexagesimal: since S is the 19 th letter of the alphabet (value 9+19 = 28) and N is the 14 th (becoming 40 th under this scheme of case sensitivity, value 9+40 = 49), and 28 × 60 1 + 49 × 60 0 = 1729.

The sexagesimal representation of a real number ${\displaystyle \scriptstyle n\,}$ corresponds to the series

${\displaystyle n=\sum _{i=-\infty }^{k}d_{i}\,60^{i}=\left\{\sum _{i=-\infty }^{-1}d_{i}\,60^{i}\right\}+\left\{\sum _{i=0}^{k}d_{i}\,60^{i}\right\},\,}$

with sexagesimal digits

${\displaystyle d_{i}\,\in \,\{{\rm {0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwx}}\}_{60}\equiv \{0,\,...,\,59\}_{10},\,}$

where the first term is improperly called the fractional part (it is a fraction only for rational numbers) and the second term is the integer part.

## Sequences

A159991 Powers of 60.

{1, 60, 3600, 216000, 12960000, 777600000, 46656000000, 2799360000000, 167961600000000, 10077696000000000, 604661760000000000, 36279705600000000000, 2176782336000000000000, ...}

A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits. (Note that the Babylonians didn't have a zero, they used extra spacing instead.)

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, ...}

A060707 Base-60 (Babylonian or sexagesimal) expansion of pi.

{3, 8, 29, 44, 0, 47, 25, 53, 7, 24, 57, 36, 17, 43, 4, 29, 7, 10, 3, 41, 17, 52, 36, 12, 14, 36, 44, 51, 50, 15, 33, 7, 23, 59, 9, 13, 48, 22, 12, 21, 45, 22, 56, 47, 39, 44, ...}

A091649 Base-60 (Babylonian or sexagesimal) expansion of 2 pi.

{6, 16, 59, 28, 1, 34, 51, 46, 14, 49, 55, 12, 35, 26, 8, 58, 14, 20, 7, 22, 35, 45, 12, 24, 29, 13, 29, 43, 40, 31, 6, 14, 47, 58, 18, 27, 36, 44, 24, 43, 30, 45, 53, 35, 19, ...}

A125628 Version of sexagesimal expansion of 2 pi given by the Persian mathematician Al-Kashi in the 15th Century.

{6, 16, 59, 28, 1, 34, 51, 46, 14, 50}

A070197 Base-60 (or sexagesimal or Babylonian) expansion of sqrt(2). (Pythagoras' constant)

{1, 24, 51, 10, 7, 46, 6, 4, 44, 50, 28, 51, 20, 34, 26, 20, 4, 31, 2, 38, 30, 53, 27, 38, 34, 5, 46, 18, 24, 29, 40, 16, 7, 16, 8, 56, 52, 55, 33, 23, 4, 47, 56, 56, 45, 38, ...}

A091720 Babylonian sexagesimal (base 60) expansion of 1/7.

A091721 Babylonian sexagesimal (base 60) expansion of 1/11.

A091722 Babylonian sexagesimal (base 60) expansion of 1/13.

Standard positional numeral systems
Base Latin (-ary)
(Cardinals)
Latin (-al)
(Ordinals)
9 Nonary, Novary Nonal, Noval   Enneadic
100 Centenary Centesimal, Centimal Centimal Hecatontadic
200 Ducenary, Bicentenary Dugentesimal, Bicentimal Dyacentesimal, Dyacentimal Dyahecatontadic
300 Trecenary, Tercentenary Tregentesimal, Tercentimal Tricentesimal, Tricentimal Triahecatontadic