Sexagesimal

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Sexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, was transmitted to the Babylonians, and is still used in modified form nowadays for measuring time, angles, and geographic coordinates. Sexagesimal as used in ancient Mesopotamia was not a pure base 60 system, in the sense that they didn't have 60 individual digits for their place-value notation. Instead, their cuneiform digits used ten as a sub-base in the fashion of a sign-value notation: a digit was composed of a number of narrow wedge-shaped marks representing units up to nine (Y, YY, YYY, YYYY, ... YYYYYYYYY) and a number of wide wedge-shaped marks representing tens up to five (<, <<, <<<, <<<<, <<<<<); the value of the digit was the sum of the values of its component parts, which is similar to how the Maya expressed their vigesimal digits using five as a sub-base (see Maya numerals). The article on Babylonian numerals shows these cuneiform digits for 1 through 60. In this article places are represented in modern decimal, except where otherwise noted (for example, "10" means ten and "60" means sixty).

1 Sexagesimal in Babylonia
2 Usage
- 2.1 Pop Culture
3 Fractions
4 Examples
5 See also
6 References
7 External link

[edit] 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 units and five for tens, 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

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v • d • e

Because there was no symbol for zero with either the Sumerians or the early 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.

[edit] 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.

One hour of time is divided into 60 minutes, and one minute is divided into 60 seconds. Parts of seconds are measured using the decimal system, however.

Similarly, the fundamental unit of angular measure 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.

[edit] Pop Culture

In Stel Pavlou's novel Decipher, this number is the center of focus, as the bucky ball Carbon element is used in the book to store data, and only base 60 proved able to be successfully understood by the computer used.

[edit] Fractions

The sexagesimal system is quite good for forming fractions of regular numbers (';' is the sexagesimal point and ',' separates sexagesimal positions):

Fraction	Sexagesimal representation
1/2	0;30
1/3	0;20
1/4	0;15
1/5	0;12
1/6	0;10
1/8	0;7,30
1/9	0;6,40
1/10	0;6
1/12	0;5
1/15	0;4
1/16	0;3,45
1/18	0;3,20
1/20	0;3
1/30	0;2
1/40	0;1,30
1/50	0;1,12
1/1:00	0;1 (1/60 in decimal)

However numbers that are not regular form more complicated repeating fractions. For example:

1/7 = 0;8,34,17,8,34,17, recurring

The fact that the adjacent numbers to 60, 59 and 61, are both prime implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as denominators, and that other non-regular primes have fractions that repeat with a longer period.

[edit] 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;24,51,10,7,46,6,4,44...,

The length of the tropical year in Neo-Babylonian astronomy, (see Hipparchus):

365.24579... days = 6,5;14,44,51 days ( = 6×60 ＋ 5 + 14/60 + 44/60² + 51/60³),

(The average length of a year in the Gregorian calendar is exactly 6,05;14,33 in sexagesimal notation.)

The value of π used by Ptolemy:

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

[edit] See also

[edit] 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.