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. Sexagesimal was not a pure base 60 system, because that would require 60 different numeric signs. Instead, sexagesimal was a Mixed radix system in which base ten and base six alternated. The units digits were in base ten (Y, YY, YYY, YYYY, ... YYYYYYYYY) and the tens digits were in base six (<, <<, <<<, <<<<, <<<<<) meaning (10, 20, 30, 40, 50). The sixtys were in base ten (60, 120, 180, ... 540) and the six-hundreds were in base six (600, 1200, 1800, 2400, 3000). The 3600 order was in base ten (3600, 7200, 10800, ... 32400). See Babylonian numerals for the cuneiform signs for 1 through 60.

In this article places are represented in modern decimal, except where otherwise noted. For example, 10 means ten, 60 means sixty, but the Babylonians wrote sexagesimal numbers using a mixture of place-value notation and additive sign-value notation.

Sexagesimal in Babylonia

and BlackDot.svg thus:

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 6.

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, geographic coordinates, and time.

The fundamental unit of time is the hour, of which there are 24 in a day. The secondary unit is minutes, of which there are 60 in one hour, and the tertiary unit is seconds, of which there are 60 in one minute. Parts of seconds are measured using the decimal system.

Similarly, the fundamental unit of angle is the degree, of which there are 360 in a circle. There are 60 minutes of arc in a degree, and 60 seconds of arc in a minute.

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 neighbors of 60 (i.e. 59 and 61) are prime numbers.

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

Examples

The length of a diagonal or a square root in a square of side a = 1, (YBC 7289 clay tablet):

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

\sqrt{2}

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 = 6,5;14,44,51^d ( = 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;14,33^d in sexagesimal notation.)

The value of π used by Ptolemy:

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

References

Georges Ifrah. The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, 1999. ISBN 0-471-37568-3
Hans J. Nissen, P. Damerow, R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993, ISBN 0-226-58659-6.

External link

Extensive page on Base-sixty

Positional numeral systems