Sexagesimal numbering is a numeral system with number 60 as the base. It originated with the ancient Babylonians: see Babylonian numerals. It was later used in its more modern form by Arabs during the Umayyad caliphate.

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

Unlike most other numeral systems, sexagesimal is not used so much as a means of general computation or logic, but is used both in measuring angles (see trigonometry) 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-based system.

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 1/20 = 0.03 1/30 = 0.02 1/40 = 0.01:30 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: recurrring 

Examples

• The length of a diagonal or a square root in a square of side a = 1, (YBC 7289 clay tablet (http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html)):

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 is 1;24,51,10,07,46,06,04,44,...,

• The length of the tropical year in Neo_Babylonian astronomy, (see Hipparchus):

365.24579... ^d = 365^d;14,44,51 ( = 365^d + 14/60 + 44/60² + 51/60³),

• The value of π used by Ptolemy:

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

See also

• base 20
• base 16
• base 12
• base 10
• base 8
• base 6
• base 3
• base 2
• latitude