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. Base-60 number systems have also been used in some other cultures, for instance the Ekagi of Western New Guinea (Lean 1992).
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).
The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, of which 2, 3, and 5 are prime. With so many factors, many simple fractions of sexagesimal numbers are simple. For example, an hour can be divided evenly into segments of any of twelve lengths: 60 minutes, 30 minutes, 20 minutes, etc.
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[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 Image:GrayDot.svg, Image:BlackDot.svg and Image:RedDot.svg thus:
| Numeral systems by culture | |
|---|---|
| Hindu-Arabic numerals | |
| Western Arabic Eastern Arabic Khmer | Indian family Brahmi Thai |
| East Asian numerals | |
| Chinese Counting rods | Japanese Korean |
| Alphabetic numerals | |
| Abjad Armenian Cyrillic Ge'ez | Hebrew Ionian/Greek Āryabhaṭa |
| Other systems | |
| Attic Babylonian Egyptian Etruscan | Mayan Roman Urnfield |
| List of numeral system topics | |
| Positional systems by base | |
| Decimal (10) | |
| 2, 4, 8, 16, 32, 64 | |
| 3, 9, 12, 24, 30, 36, 60, more… | |
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] Popular 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. At least one popular book[1] uses the spelling "sexigesimal" instead of "sexagesimal," with the latter being the more common spelling of the word.
[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.
In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: prime or primus, seconde or secundus, tierce, quatre, quinte, etc. To this day we call the second-order part of an hour or of a degree a "second". In the 1700s, at least, 1/60 of a second was called a "tierce" or "third". [1], [2]
[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 <math>\sqrt{2}</math> 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 6,5;14,33 in sexagesimal notation.)
- 3.141666... ≈ 377/120 = 3;8,30 = ( 3 + 8/60 + 30/602 ).
[edit] See also
[edit] References
- ^ Mlodinow, Leonard: "Euclid's Window", page 10. The Free Press, 2001
- Ifrah, Georges (1999), The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley, ISBN 0-471-37568-3.
- Nissen, Hans J.; Damerow, P. & Englund, R. (1993), Archaic Bookkeeping, University of Chicago Press, ISBN 0-226-58659-6.
- Lean, Glendon Angove (1992), Counting Systems of Papua New Guinea and Oceania, Ph.D. thesis, Papua New Guinea University of Technology, <http://www.uog.ac.pg/glec/thesis/thesis.htm>. See especially chapter 4.
[edit] External links
- Extensive page on Base-sixtyca:Sistema sexagesimal
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