Sexagesimal numeral system

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The sexagesimal numeral system 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 $\scriptstyle n\,$ corresponds to the series

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

with sexagesimal digits

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

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.

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

External links

• Number Bases Conversion from ConvertXY.com
• Jack Sanders-Reed, Base Conversion (Java applet calculator capable of converting a number from/to any base 2-36.)
• Base Conversion (Convert to/from base two through base sixteen.)
• Base Convert (Convert to/from any integer base greater than 1 or Roman numerals.)