# Sexagesimal

The sexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, and was transmitted to the Babylonians: see Babylonian numerals.

Usually, the sexagesimal uses ten numerals (from 0 to 9) according to decimal place system as the sub-base.

In this article places are based on decimal, except where otherwise noted. For example, 10 means ten, 60 means sixty.

## Sexagesimal in Babylonia

The Sumero-Babylonian version used a digit to represent "one" and another digit to represent "ten", and repeated the symbols in groups up to nine for the former and five for the latter, then used place-position shifting to the left for each power of sixty, with a larger space between one power of sixty and the next -- this may be represented schematically here by using / and . thus:

/  //  ///  ///  ///  ///  ///  ///  ///   .  ./ .// ./// .. ...  ...///   /   / /
/   //  ///  ///  ///  ///                      ..   ..///
/   //  ///                             ///

1   2    3    4    5    6    7    8    9  10  11  12   13 20  50      59  60    61


Because there was no symbol for zero with either the Sumerians or the earlier Babylonians, it is not always immediately obvious how a number should be interpreted, and the true value must sometimes be determined by the context; later Babylonian texts used a dot to represent zero.

It was later used in its more modern form by Arabs during the Umayyad caliphate.

## Usage

60 (sexagesimal) is the product of 3, 4, and 5. 3 is a divisor of 12 (duodecimal), 4 is a common divisor of 12 (duodecimal) and 20 (vigesimal), 5 is a common divisor of 10 (decimal) and 20 (vigesimal).

Base-sixty has the advantage that its base has a large number of conveniently sized divisors {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, facilitating calculations with vulgar fractions. Note that 60 is the smallest number divisible by every number from 1 to 5.

Unlike most other numeral systems, sexagesimal is not used so much as a means of general computation or logic, but is used in measuring angles, time and geographic coordinates. The standard unit in sexagesimal is the degree, of which there are 360. The secondary unit is the minute, of which there are 60 minutes in one degree. The tertiary unit is the second, of which there are 60 seconds in one minute.

The modern use of sexagesimal corresponds very closely with the modern measurement of time, in which there are 24 hours in a day, 60 minutes in one hour, and 60 seconds in one minute. The modern measurement of time roughly corresponds to the rotation (days) and revolution (years) of the Earth. Units that are smaller than one second are measured using a decimal system.

In the Chinese calendar, a sexagenary cycle is commonly used.

## Fractions

The sexagesimal system is quite good for forming fractions:

 1/2 = 0.30
1/3 = 0.20
1/4 = 0.15
1/5 = 0.12
1/6 = 0.10
1/8 = 0.07:30
1/9 = 0.06:40
1/10 = 0.06
1/12 = 0.05
1/15 = 0.04
1/16 = 0.03:45
1/18 = 0.03:20
1/20 = 0.03
1/30 = 0.02
1/40 = 0.01:30
1/50 = 0.01:12
1/1:00 = 0.01 (1/60 in decimal)


but is not very good for simple repeating fractions, because both the neighbours of 60 (i.e. 59 and 61) are prime numbers.

 1/7 = 0.08:34:17:08:34:17: recurring


## Examples

1.414212... = 30547/21600 = 1.24:51:10 (sexagesimal = 1 + 24/60 + 51/602 + 10/603), a constant used by Babylonian mathematicians in the Old Babylonian Period (1900 BC - 1650 BC), the actual value for $\displaystyle \sqrt{2}$ is 1.24:51:10:07:46:06:04:44...,
365.24579... d = 6,5;14,44,51d ( = 6×60 ＋ 5 + 14/60 + 44/602 + 51/603),

(Note that the average length of a year in the Gregorian calendar is exactly 6,5;14,33d in sexagesimal notation.)

3.141666... = 377/120 = 3.8:30 ( = 3 + 8/60 + 30/602 ).