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Sexagesimal numeral system

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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 corresponds to the series

with sexagesimal digits

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.

See also

Standard positional numeral systems
Base Latin (-ary)
(Cardinals)
Latin (-al)
(Ordinals)
Greek (-al) Greek (-adic) Usage
1 Unary, Primary Primal   Monadic, Henadic  
2 Binary, Secondary Dual   Dyadic  
3 Ternary, Tertiary Tertial   Triadic  
4 Quaternary Quartal   Tetradic  
5 Quinary Quintal Pental Pentadic  
6 Senary Sextal Hexal Hexadic  
7 Septenary Septimal Heptal Heptadic  
8 Octonary Octonal Octal Octadic  
9 Nonary, Novary Nonal, Noval   Enneadic  
10 Denary Decimal Decimal Decadic  
11 Undenary, Unodenary Undecimal, Unodecimal Henadecimal Hendecadic  
12 Duodenary, Dozenary Duodecimal, Dozenal Dyadecimal Dyadecadic  
13 Tredenary, Triodenary Tredecimal, Triodecimal Tridecimal Triadecadic  
14 Quadrodenary, Quattuordenary Quadrodecimal, Quattuordecimal Tetradecimal Tetradecadic  
15 Quindenary Quindecimal Pentadecimal Pentadecadic  
16 Sedenary, Sexadenary Sedecimal, Sexadecimal Hexadecimal Hexadecadic  
17 Septendenary Septendecimal Heptadecimal Heptadecadic  
18 Octodenary Octodecimal Octadecimal Octadecadic  
19 Nonadenary Nonadecimal, Novodecimal Enneadecimal Enneadecadic  
20 Vigenary Vigesimal   Icosadic  
24   Quadrivigesimal   Tetraicosadic  
26   Sexavigesimal   Hexaicosadic  
27   Septemvigesimal   Heptaicosadic  
30 Tricenary Trigesimal   Triacontadic  
32   Duotrigesimal   Dyatriacontadic  
36   Sexatrigesimal   Hexatriacontadic  
40 Quadragenary Quadragesimal Tetragesimal Tetracontadic  
50 Quincagenary Quincagesimal Pentagesimal Pentacontadic  
60 Sexagenary Sexagesimal Hexagesimal Hexacontadic  
64 Quadrisexagenary Quadrisexagesimal Tetrahexagesimal Tetrahexacontadic Base64 encoding
70 Septuagenary Septuagesimal Heptagesimal Heptacontadic  
80 Octogenary Octogesimal Octagesimal Octacontadic  
90 Nonagenary Nonagesimal Enneagesimal Enneacontadic  
100 Centenary Centesimal, Centimal Centimal Hecatontadic  
200 Ducenary, Bicentenary Dugentesimal, Bicentimal Dyacentesimal, Dyacentimal Dyahecatontadic  
300 Trecenary, Tercentenary Tregentesimal, Tercentimal Tricentesimal, Tricentimal Triahecatontadic  
400 Quadringenary, Quadricentenary Quadringentesimal, Quattrocentimal Tetracentesimal, Tetracentimal Tetrahecatontadic  
500 Quingenary, Quinquocentenary Quingentesimal, Quincentimal Pentacentesimal, Pentacentimal Pentahecatontadic  
600 Sexingenary, Sexcentenary Sexingesimal, Sescentimal Hexacentesimal, Hexacentimal Hexahecatontadic  
700 Septingenary, Septcentenary Septingentesimal, Septuacentimal Heptacentesimal, Heptacentimal Heptahecatontadic  
800 Octingenary, Octocentenary Octingentesimal, Octocentimal Octacentesimal, Octacentimal Octahecatontadic  
900 Nongenary, Nonacentenary Noningentesimal, Novacentimal Enneacentesimal, Enneacentimal Enneahecatontadic  
1000 Millenary Millesimal   Chiliadic  
10000       Myriadic  

Notes

  1. Weisstein, Eric W., Sexagesimal, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Sexagesimal.html]

External links